THERMAL FLOW SIMULATION FOR CASTING/MOLDING PROCESSES

Abstract: A method for simulating the behavior of an incompressible, viscous fluid in a casting/molding process (102). The method uses classical fluid dynamic equations and control volume-finite element techniques to solve for the momentum and energy of the fluid (102). The method optionally takes into account, during the simulation, fluid flow in the shot sleeve (109), formation of shrinkage porosity, heat transfer between the die and the heat transfer fluid, cooling effect of the spray lubricant on the die, and formation of the mend line.

Description: THERMAL FLOW SIMULATION FOR CASTING/MOLDING PROCESSES CROSS REFERENCE TO RELATED APPLICATION The present application is a continuation-in-part application of a commonly owned, co-pending U.S. Patent Application, entitled THERMAL FLOW SIMULATION FOR CASTING/MOLDING PROCESSES, Serial No. 09/610,390, filed on July 1, 2000 by the same inventive entity. The referenced application is incorporated herein by reference in its entirety. FIELD OF THE INVENTION This invention relates to a method for simulating the behavior of a fluid or melt during a die casting or mold filling process, in particular, to a numerical method for simulating incompressible, viscous melt flow and heat transfer in a die. BACKGROUND OF THE INVENTION In manufacturing casting/molding processes, for product quality of the molded article, it is of importance to optimize the mold design and the processing conditions. Conventionally, such optimization is attained after many trial and error experiments which are repeated on the basis of the rule of thumb of an expert. With the advance in computer technology, it has become possible to analyze flow and heat transfer behavior of a material during a casting molding process by computer simulation. When realistically modeled, the results from such simulation may be used to predict the casting behavior, thereby improving the efficiency of product and mold-design, and optimization of processing conditions. Conventional methods for simulating fluid flow in a casting/molding process generally involve applying the equations of motion, continuity and energy to a discretized model of the molded part and performing numerical calculations thereof to obtain velocity, pressure, and temperature distributions along the processing line. In these simulations, the thermal physical properties of the material, the operational characteristic such as temperature of the material, temperature of the mold and injection speed, etc., are required input for the simulation. These inputs are generally assumptions made based on experimental know-how obtained from repeated comparisons between the analytical results and the actual moldings. The conventional models generally provide no means for determining whether or not input conditions are appropriate. It is therefore necessary to supply realistic boundary conditions before an accurate simulation can be achieved. Conventional fluid flow models lack many refinements which would enable better estimation of the physical boundary conditions, and hence often fail to achieve an accurate simulation. One refinement which would be desirable, is incorporating the flow in the shot sleeve into the model. In a traditional die-casting process, molten material is delivered into mold cavity from an injection unit, which typically includes a plunger (or ram), shot sleeve, gate and runner. Pressure is applied via the plunger to inject the molten material from the shot sleeve into the runner through a gate. As the plunger and the melt moves, the interface between them become a moving discontinuities, which often cause instability in numerical models. Therefore, many prior art models start the simulation at the gate entrance and make the assumption that the temperature and pressure or velocity applied by the ram are uniform and constant across the gate surface. In practice, the temperature of the melt in the shot sleeve is neither uniform nor constant as the material loses heat to the shot sleeve and the plunger. Likewise the velocity and pressure field in the shot sleeve could be quite complicated. More importantly, the pressure applied by the plunger and thickness and temperature of the biscuit remains in the shot sleeve after the cavity is filled have a major impact on the effectiveness of pressurization to suppress the formation of porosity in the molded/casted parts. One of the objectives of this invention is to provide an effective and reliable algorithm to incorporate the thermal flow in the short sleeve in casting/molding simulations. Another improvement is to incorporate the heat exchange between the die and the heat-transfer fluid (HTF) into the model. Mold temperature is a common process parameter of all metal casting or plastic injection molding processes. In order to maintain a steady die temperature between injection cycles, HTF lines is often embedded in the body of the die. Heated oil or water can be used to pre-heat the die to shorten the start-up time and to reduce the corresponding scrap. During continuous operation, the temperature of the HTF can be adjusted to maintain the cyclic thermal balance in the die. Many attempts have been made to incorporate the heat transfer between the die and the HTF in the process simulation. The most vigorous and resource-demanding approach is to use full 3D models to simulate the flow and convection heat transfer of the HTF inside the duct. In order to reduce the computational time, most of the existing models neglect the flow or temperature change of the HTF in the die. Instead, the user is required to select the element surfaces of the heat transfer lines and to assign a HTF temperature and a heat transfer coefficient, as thermal boundary conditions. Since the HTF's viscosity changes with its temperature, and the HTF's temperature varies through the entire loop, the Reynolds number and Prandtl number are not constant along the HTF lines. It is therefore difficult for the user to determine the appropriate heat transfer coefficient. Therefore, it is desirable to have an efficient method which updates the HTF temperature accurately without heavy computation and determine the heat transfer coefficient of the HTF automatically such that the heat flux from the die to the HTF may be accurately predicted. A method for predicting the formation of shrinkage porosity would also be a desirable feature to be incorporated in a metal casting simulation model. Shrinkage porosity formation is a common problem for a metal casting process, e.g. semi-solid forming. When molten metal fills the cavity in a die which is at a lower temperature than the molten metal, the molten metal loses its thermal energy to the die and begins to solidify. As the liquid phase changes to solid, its density increases. If there is no additional material fed into the cavity, the net volume of the metal will become less than that of the cavity and induce the formation of shrinkage porosity. Depending on its size and distribution, shrinkage porosity could significantly compromise the mechanical properties of a product, especially its elongation and fatigue strength. For pressure tight products, shrinkage porosity may randomly form multiple chained paths and cause leaking. Accurate predictions of shrinkage porosity are advantageous, particularly when designing a structural or leak-tight product, e.g. control arm and knuckle, air-conditioner housing and fuel-injection rail. So far, the state-of-the-art die casting simulation programs can only display potential locations of shrinkage porosity based on the hot spots (non-solidified metal with higher temperature surrounded by solidified colder metal) within the cast's volume. In addition to temperature distribution, some of the programs also incorporate the metal's cooling rate to predict shrinkage porosity. However, the formation of shrinkage porosity is actually coupled not just with the thermal factors but also with the flow and pressurization. Ji the molten material in the die cavity is in direct communication with a pressurized molten material source, e.g. the remaining molten material in the shot sleeve, additional material may be pushed into the die cavity to compensate for the volume shrinkage and to suppress the formation of shrinkage porosity. The required pressure depends on the viscosity of the solidifying material, the molten material's temperature and the die cavity's geometry. For accurate predictions and hence prevention of the formation of shrinkage porosity, a model that simulates the pressurized thermal flow with porosity incorporated in the mass conservation equation would be required. Another feature that is important to any die-casting or mold-filling process is monitoring the cooling of the die or mold when lubricant is sprayed. Generally in a pressure casting process, after the metal solidifies in the cavity, the die opens to eject the part. A water-based lubricant is often sprayed on the die walls for die release and prevention of drag mark or soldering. As the lubricant typically contains 98% water, it removes significant amount of the thermal energy transferred to the die from the alloy. In order to predict the die temperature and its heat transfer with the alloy, the cooling effects of the lubricant spray in the model should be incorporated. Generally, there may be multiple moving or stationary nozzles spraying lubricants from different angles or distances. Sometimes, these nozzles may move back and forth to spray lubricant more uniformly. Furthermore, depending on the type of nozzle, spray pressure and die temperature, the resulting distribution of cooling effects is not uniform. As a result, it is very difficult for a user to define the corresponding cooling coefficient on the die or mold surface, especially for a die with complex cavity shape. In some of the existing programs, the user has to manually select and assign cooling coefficients on those elements surfaces that may be covered by the spray. In cases where that the nozzle moves, it would take too much time to set up the cooling coefficients. A model that computes the cooling efficiency from the geometry of the spray nozzle and its motion would be desirable. A model to predict the location of a mend line and trapped air would also be a desirable feature for incorporation in a casting/molding simulation. When molten material fills a cavity, the melt front may split, e.g. to go around a core or a hole, and re-join. As the melt front is in contact with the die steel and air, it may become too cold and/or contaminated by residual lubricant or moisture. In addition, if the die is inappropriately designed, the cavity may not be filled progressively and air may be trapped inside the molten material. As a result, wherever melt fronts meet, the bonding on the interface may be weakened due to pre-mature solidification, trapped air or impurities. Although the user can display the filling pattern from various angles to search for mend lines, it is very time consuming and not reliable, especially if the part is large and complicated. Furthermore, after the melt fronts meet, they could continue to move if the cavity was not completely filled at that time. As a result, the final position of the weakened bonding interface may be different from the original mending location. In addition, the strength of bonding could be improved if there is a vigorous mixing of the material across the interface. It would be advantageous in the design stage to be able to identify these potential casting defects quantitatively so that preventive design can be incorporated early in the design stage. Otherwise, the processing window would become very tight and difficult to maintain consistent quality. Therefore, there is a need for a new method which incorporates the aforementioned refinements such that the behavior of an incompressible, viscous fluid flow in a casting/molding process can be accurately modeled by computer simulations. SUMMARY OF THE INVENTION Accordingly, an object of the present invention is to provide an improved method for simulating incompressible, viscous melt flow in a casting molding process. Related objects and advantages of the present invention will be apparent from the following description. The present invention provides a method for simulating thermal flow during a die casting or mold filling process, which satisfies the Navier-Stokes equations and the energy equation. This method provides five flow and thermal modules, namely flow in a shot-sleeve, heat exchange in heat-transfer fluid line, die cooling by lubricant, shrinkage porosity formation, and mend line formation, each of which represents an optional component that could be incorporated separately or jointly in a casting/molding process. The incorporation of these modules provides better prediction of the boundary conditions and hence more accurate simulation. The modules may be incorporated in the overall casting model in any combination. One aspect of the invention is a method for simulating thermal flows for casting/molding processes, including: (a) selecting simulation modules for incorporation in an overall simulation sequence, said modules including a Flow in Shot Sleeve Module, a Shrinkage Porosity Prediction Module, a Heat Transfer Fluid Line Module, and a Die Lubricant Cooling Module; discretizing flow domain into elements having nodes and fluxing surfaces, inputting initial parameter value and process conditions; (b) incrementing the time by an incrementing time step; (c) updating material balances; (d) forming and solving momentum equations, and updating velocity and pressure fields; (e) forming and solving energy equations, and updating temperature field; (f) checking if parameters have converged, returning to step (d) if they have not; (g) determining whether an end-simulation event has occurred, returning to step (c) if it has not; and (h) deciding whether to execute post- processor simulation modules, said post-processor simulation modules including a Mend Line Prediction Module. BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a schematic diagram of a machine to implement the present invention. FIG. 2 is a flowchart of an embodiment of the overall solution sequence of the method of the present invention. FIG. 3 is a schematic drawing of a triangular finite element suitable for modeling a two-dimensional fluid flow of the method of the present invention. FIG. 4 is a schematic drawing showing a two-dimension nodal control volume. FIG. 5 is a schematic drawing of a tetrahedral finite element suitable for modeling a three dimensional fluid flow of the method of the present invention. FIG. 6 is an example of a two-dimensional flow domain showing a boundary of a computational domain. FIG. 7 is a flowchart of an embodiment of the Flow in the Shot Sleeve Module of the method of the present invention. FIG. 8 is a flowchart of an embodiment of the Shrinkage Porosity Prediction Module of the method of the present invention. FIG. 9 is a schematic drawing of the heat transfer fluid line showing the transfer line segments in relation to the mold faces. FIG. 10 is a flowchart of an embodiment of the Heat Transfer Fluid Line Module of the method of the present invention. FIG. 11 is a flowchart of an embodiment of the Die Lubricant Cooling Module of the method of the present invention. FIG. 12 is a flowchart of an embodiment of the Mend Line Prediction Module of the method of the present invention. DESCRIPTION OF THE PREFERRED EMBODIMENT For the purposes of promoting an understanding of the principles of the invention, reference will now be made to the embodiment illustrated in the drawings and specific language will be used to describe the same. It will nevertheless be understood that no limitation of the scope of the invention is thereby intended. Any alterations and further modifications in the illustrated device, and any further applications of the principles of the invention as illustrated therein being contemplated as would normally occur to one skilled in the art to which the invention relates are also included. The illustrated embodiment of the present invention is a numerical method for computer simulation of the behavior of an incompressible, viscous fluid or melt during a casting/ molding process. The illustrated method is based on classical fluid dynamic models, and numerically (through the use of a computation device, generally a computer) obtains solutions to the governing equations at user defined time intervals. The illustrated method has general application to all thermal flows in metal die casting and polymer molding processes. The present method improves upon prior art methods by providing five novel modules: a Flow in the Shot Sleeve Module, a Heat Transfer Line in Die Module, a Die Lubricant Cooling Module, a Shrinkage Porosity Prediction Module, and a Mend Line Prediction Module, each of which simulates an aspect of the casting/molding process. These modules provide realistic boundary conditions for solving the governing fluid dynamic equations. Depending on the objective of the thermal flow simulation, a user optionally selects to incorporate all or any number of the modules into the simulation. The sequential operations of each module are described in individual flow charts, separate from the overall simulation sequence. In these flow charts, the modules are depicted as complete sub-routines, mainly for illustrative purposes. However, it should be apparent to those skilled in the art that in an actual simulation where these modules are incorporated, the operations of the modules are integrated, in varying degrees, into the overall simulation sequence, and may become indistinguishable. The illustrated embodiments of the present invention are designed to operate within a processor based computation device. Typically, the logic routines of the present method resides on a storage medium, e.g. a hard drive, an optical disk, magnetic tapes or any state of art storage medium which is accessible by the processor. Referring now to FIG. 1, there is shown a typical processor system which is capable of carrying out the method of the invention. A conventional computer system 2, including a processor 4 having access to read only memories (ROMs) and random access memories (RAMs), operatively connected by cables and other means to input and output devices. As illustrated, these input and output devices may include a display 5, a keyboard 6, a mouse 7, a printer 8, an input/output storage device 9, or any combinations thereof. Processor 4, in cooperation with an operating system, executes the logic routines of the present method, receives and stores data from the input devices, performs computations, and transmits data to the output devices. Alternatively, the executable program, the input data and the output data could be stored in the local system or in a remote system that can be accessed through network. While conventional computer systems are illustrated, other systems may be substituted without departing from the invention. Specifically, the illustrated embodiment of the method operates within a Silicon Graphic System using UNTX operating system. However, other systems having similar computation capability, memory capacity and graphic displays can also be used. An embodiment of the method of the present invention is developed based on a finite-difference or a finite-element procedure where the flow domain is subdivided into a grid or a mesh of finite elements having nodes (or grid points), where, variables, e.g. velocity, pressure and melt-front location parameter are defined. These variables includes scalar, e.g. pressure and temperature, and vectors, e.g. velocity in two/three dimensional problems. Following traditional finite-difference or finite-element theories, the variables can be interpolates using its nodal values linearly or a shape function. The governing equations are discretized to form linear algebraic equations following several conventional finite difference (Numerical Heat Transfer and Fluid Flow by S. V. Patankar, Hemisphere Publishing Corporation) or finite element procedures, e.g. Ritz, Galerkin, least-squares, collocation or, in general, weighted-residual finite-element methods. (An Introduction to the Finite Element Method by J. N. Reddy, McGraw-Hill International Book Company) or Control- Volume Based Finite-Element Formulation of the Heat Conduction Equation", Progress in Astronomical and Aeronautical Engineering, Vol. 86, pp. 305-327, 1983). There are many ways to solve the resulting discretized momentum and mass-conservation equations. Any solver that could generate converged solution within acceptable error is applicable with the proposed method. In the preferred embodiment, the discretized equations are solved by the Equal-Order method. The Equal Order method introduces a Gauss point velocity field and enables pressure and velocity be evaluated at every nodal point. The method provides accurate solutions to the velocity and pressure of the fluid, such that the mass conservation of the fluid can be satisfied to the truncation limit of the computational device (machine round-off levels). The equal order method will be discussed in detailed later. While the equal order method is preferred, any methods of solving the momentum equation and capable of conserving mass to machine round off level may also be used. In this method the governing equations (1) - (3) can be expressed as: Conservation of mass: (1) Conservation of momentum: p ^—+ p (V • V )V = -V P + pg + μ V 2 V d τ (2) Energy Equation P C P ^∑_+ p C P (V V T ) = V (kV T ) + q"+Φ d τ (3) where: τ= time p = density P = pressure V = the velocity vector μ = viscosity g = gravity components T = temperature C p = heat capacity Φ = viscous heating source term q = heat flux OVERALL SIMULATION SEQUENCE Referring to FIG. 2 which shows the flowchart of the overall simulation sequencelOO, according to an embodiment of the present invention. At stage 101, the flow domain is discretized into a mesh of element. The matrix representing the mesh and initial parameter values and other boundary conditions are input. For modeling the flow domain, a finite difference grid or a finite element mesh may be used. A finite-difference grid discretizes a space with regular lines that are parallel to the coordinates and hence may not have the flexibility for modeling of a complex domain. An irregular finite element mesh is therefore preferred. A finite element mesh may be created by any conventional procedures. Such suitable procedures have been disclosed, for example, in U.S. Patent No. 5,010,501 issued on April 24, 1991 to Arakawa and entitled "Three-Dimensional Geometry Processing Method and Apparatus Therefor", and U.S. Patent 5,553,206 issued on September 3, 1996 to Meshkat and entitled "Method and System for Producing Mesh Representations of Objects". Both the finite difference and finite element can be one, two or three- dimensional depending on the flow domain being modeled. For modeling a two dimensional flow problem, a mesh of triangular finite elements similar to that shown in FIG. 3 and 4 is preferred for its geometric flexibility. However, a quadrilateral element is also applicable. Referring now to FIG. 3, the triangle finite element 10 includes three vertex nodes 12. The variable parameters of the fluid, i.e. temperature, velocity and pressures are evaluated at nodes 12 of element 10. Particular of the Equal-Order method, a Gauss-point velocity, V g and a Gauss- point pressure gradient VP g are defined inside each element so that they have the same order and are computed from the variables at the nodal points 12. Finite element 10 can be divided into three sub-control volumes (SCVs) 18 by joining the mid-side points 14 to the centroid 16. Each of sub-control volumes 18 is bounded by four fluxing surfaces 30 where material can flow from one sub- control volume to another. These sub-control volumes 18 surrounding a vertex node 12 form the control volume (CV) for that node. For example, as shown in FIG. 4, vertex node 22 is surrounded by five sub-control volumes 23-27 and the shaded area represents control volume 20 of vertex node 22. While it is illustrated that control volume 20 is composed of five sub-control volumes 23-27, it is possible that only some of the sub-control volumes are filled or partially-filled by the fluid. Therefore, only those filled and partially-filled control volumes are included in the computational domain. For a given mesh, defining the filling status in each SCV can provide more flexibility and better resolution to the predicted moving free surfaces than defining filling status only in the whole control volumes. However, defining filling status in the whole control volume simplifies the program' s structure. For modeling a three-dimensional problem, a tetrahedral element is preferred over hexahedronal elements for the convenience of available automatic tetrahedral mesher. As shown in FIG. 4, a linear tetrahedral element 40 has four vertex nodes 42 and four surfaces 43. Finite element 40 can be divided into four sub-control volumes (SCVs) 48 by joining the surface centroid 44 (centroid at each surface of the tetrahedral element 40) to the volume centroid 46 (the centroid of the tetrahedron). Each of the sub-control volumes 48 is a hexahedron having six fluxing surfaces. While triangular and tetrahedral finite elements are illustrated and preferred, it should be understood that other geometrically shaped finite elements, a combination thereof or a finite difference method may be used. In addition, linear finite elements are illustrated for its simplicity. It is however, contemplated that the finite elements may be of any order. Preferably, a computational domain is defined to exclude unneeded variables and to reduce computational time. The boundary of the computational flow domain may be drawn according to Wang, et al., as disclosed in U.S. Patent No. 5,572,434; and the disclosure of which in expressly incorporated in its entirety. Wang draws the boundary of a computational domain according to the filling status of the element in the flow domain. Wang defines, for triangular elements as shown in FIG. 6, that an empty element 62 is an element having all of its sub- control volume empty; a filled element 64 is having all of its sub-control volumes filled, and a partially filled elements 66 is an element having at least one, but less than all of its sub-control volume filled. Further, Wang designates the nodes of the flow domain as interior or exterior nodes. If a node is surrounded by only filled sub-control volumes, it is designated as an "interior node" 67. All other nodes are "exterior nodes." These exterior nodes are further divided into several groups: exterior nodes associated with filled and partially-filled sub-control volumes are known as "free surface nodes" 68, those immediately adjacent to free surface nodes, but not themselves associated with a filled sub-control volume) are called "driven nodes" 69. The boundary 70 of a computational domain is defined as the line linking those exterior nodes which are associated with empty sub-control volumes and adjoining driven nodes. While Wang's method of defining a computational flow domain is illustrated, any other methods of defining a computational flow domain and allowing meaningful solution to be obtained may also be used without deviates from the scope of the invention. At stages 102, the user selects the thermal flow modules to incorporate into the simulation and, the process condition to be included in the simulation. The user further selects the number of cycle to simulate and the convergence criterion for the variable parameters. The user may select to include in the simulation one or any combination of the following thermal and flow modules: Flow in the Shot Sleeve 200, Shrinkage Porosity Prediction 300, Heat Transfer Fluid Line in Die 400, Die Lubricant Cooling 500, and Mend Line Prediction 600. The method then integrates the selected modules in the overall solution sequence based on the actual sequence in casting/molding process. It should be understood that, with the exception of the Mend Line Prediction module 600, these modules are not executed independently. The step sequences of the modules are intersperse between the step sequence of the overall simulation module such that the two step sequences become interdependent and indistinguishable. For each of the modules selected, specific process parameters and boundary conditions may be required. For example, ram velocity and pressure profiles are required entry for the Flow in Shot Sleeve module 200. The method prompts the user to supply the necessary conditions. However, if the user rejects not to, the method will provide default process conditions from its memory bank. The user may select the number of cycles to simulate. A cycle is a complete filling event, from supplying the shot sleeve with fresh fluid to the filling of the die cavity and the final ejection of the casted part. The method provides a counter to determine the number of elapsed cycles. The user should set the number of cycles to simulate based on the process phenomenon of interest. Generally, during start-up of a process run, it is necessary to run through a few cycles to allow the components to come up to steady state. The user may also select the criteria to determine the convergence of the variable parameters. An iterative process is said to have converged when further iterations will not improve the accuracy of the dependent variables. In practice, the iterative process is terminated when the desired accuracy is obtained. As the momentum equations and energy equation are non-linear and the fluid's properties may change with the velocity field and temperature, iterations are required until the incremental change is negligibly small. Therefore, it is preferred that the incremental difference of the variable parameter between two successive iteration be selected as the criterion for convergence. However, other suitable criterion may also be selected. Further, the user may select the tolerance for this convergence criterion. At stage 103, all the variable parameters are initialized. The method resets the cycle counter and the master clock to zero, signaling the start of the simulation. At stage 104, the method initializes the cavity variables, including the shot sleeve and the HTF line if the user has selected the Flow in Shot Sleeve and the HTF Line in Die modules. At stage 105, at each subsequent run cycle, the cycle counter advances by 1. At stage 106, for each subsequent time step, time τ, is advanced by an incremental interval AT. The user may choose to fix Aτ or allows it to vary. Preferably, AT is allowed to vary. A varying Δr permits the user to define the incremental changes of the variable parameters in each time step, such that convergence of the variable parameters throughout the simulation can be achieved with a small number of iterations. At stages 107, if the user has selected to incorporate Flow in Shot Sleeve Module 200, the method determines whether the plunger (or ram) is under pressure control or velocity control. If the plunger is under velocity control, the method determines the volume of fluid displaced by the plunger. This volume of fluids determines the mass flux across the die cavity. The method then updates the location of the plunger in preparation of a subsequent time step. If the plunger is under pressure control, the method determines the net pressure (the difference between the pushing force of the plunger and the resistance force of the fluid) the plunger exerts on the fluid and determine the velocity across the interface. If the user did not select the Flow in Shot Sleeve module 200, the method moved stage 108. At stage 108, the filling status of the sub-control volumes, the position of the melt front, the trapped air pockets and other dependable variables based on the new temperature and velocity fields are updated. The instantaneous volume of a fluid in a sub-control volume is the sum of its initial volume and the net-influx of melt from the beginning until the time step being analyzed. The mass flux is computed from the fluid density, velocity, the fluxing area and time. The amount of melt in a nodal control volume is the sum of the melt in all the attached sub-control volumes. The fill fraction of the melt can be computed by tracking each control volume. In an preferred embodiment of the method, the mass flux contributions around a nodal control volume,/ is preferably computed using a Gauss point velocity field (u g , v g , w g ) that: Jj = p{Ug(Oxj + VgCOyj + WgCDzj) (4) where J j is the mass flux around a nodal control volume j, p is density, and ω is the fluxing surface area. Computing mass flux using the Gauss point velocity field enables global mass conservation to be computed to the truncation limits of the computation device (machine round-off levels). In addition, the method also uses a weighting scheme to compute mass flux to ensure global mass balance is conserved. For mass fluxes that are filling a nodal control volume, the upstream fill fraction is used to weight the filling flux. If a mass flux is draining a nodal control volume, then the downstream fill fraction is used to weight the flux. This weighting scheme is much like the donor-acceptor algorithm used by SOLAVOF. At this stage, the position of the free surface is also located. The volume fraction of the melt,/ m , is used to infer the position of the free surface;/ m , which is defined as: Jm — ' m ' ' (5) where V and V m denote the volume of the control volume and the volume occupied by the melt, respectively. In an element where the melt front exists, the volume fraction of the elt,f m , on some of the nodes would be 1 and others less than 1. The free surface is defined as the surface where the volume fraction of melt, as interpolated from the nodes, is 0.5. At this stage, the trapped air pockets within the flow domain may also be identified. A trapped air pocket is a single or a group of empty sub-control volumes surrounded by filled or partially filled sub-control volumes. These trapped air pockets are named and are assigned atmospheric pressure as their initial pressure. The position and melt fraction of these trapped air pockets are tracked and updated with each subsequent time step. The pressure within the trapped air pocket is calculated from the ideal gas law using the total mass flux into the pocket during a specific time interval. The updated pressure becomes the boundary condition for any in-filling event at the next iteration. After updating the filling status of the nodal control volumes and determining the melt front, the program will update the boundary of the computational domain according to the definition described previously. At stage 109, the method approximate a solution to the Navier Stroke equation to obtain the values for pressure and velocity. The method forms discretized equations and solves them based on the Equal Order method. The development of the Equal-Order method is disclosed in an application entitled "Equal Order Method for Fluid Flow Simulation", the disclosure of which is hereby incorporated by reference in its entirety. The following is a brief description of the method. Considering a two-dimensional case for simplicity, the assembled momentum equations can be modeled by equations (6) and (7): d P ∑ a ij u j 4A i= ι d x (6) and (7) where the ay terms make up the influence coefficient matrix and n is the total number of fluxing faces of the neighboring sub-control volumes. Pulling the pressure gradients outside the area integrals, equations (6) and (7) are recast as equations (8) - (12) as follows: V i = V i - K d P d y (8) d P = u K (9) and where, 1 γ-ι u = ayu j a ii j ι ±≠ j i (10) (11) if = — f d , (12) The nodal hat velocities, U. and V. , are composed of nodal velocities U j and V j of neighboring sub-control volumes and contain no pressure term. The gauss point hat velocity U it g , V Λ g and the Gauss point coefficient term K g are defined as: (13) » * = ∑ »y, 2 V _ • = ∑ Vj , (14) K g — — J K J n j = i (15) where n e is the number of fluxing faces within the element. The Gauss-point velocities are expressed as: d P V g = v R - K d y (16) d P u g — u g K d x (17) In the above two-dimension expression for the Gauss point velocities, u g and v g , the pressure gradients, dPldx, dP/dy, are constant over the fluxing surface. For consistency, the terms u gt v g and K g are constant over the fluxing surface as well. By inserting the expression for the Gauss-point fluxing velocities, u g and v g , into the continuity equation (1), a Poisson equation for pressure is obtained. The pressure equation is expressed as: u V k V p = + d v d x d y (18) Integrating this equation over the area of the fluxing surface of the element, the pressure field can be expressed as: (19) Using Green's theorem, these area integrals are converted to line integrals, (20) Using the gauss point hat velocities U g , V g and K g previously defined, the Poisson pressure equation can be assembled and solved to determine the pressure field. The nodal velocities, Ui and Vj, and the gauss point velocities, U g and V g , are computed by inserting the newly derived pressure field and U. , V , K{, U g , V g and K g in the momentum equations. While the Equal Order method is shown and preferred, it should be understood that other methods which derive and solve the discretized form of the mass and momentum equation may be used without deviating from the spirit and scope of this invention. For example, the momentum equation and mass conservation equations can be combined into a set of linear matrix equation and solved by using direct solver, e.g. matrix-inverse method, or iterative solver, e.g. conjugate-gradient method. In addition, if Porosity Prediction module 300 is selected, the continuity equation is modified to include the effects of porosity expressed in terms of volumetric fraction of porosity that the apparent density of the alloy β=(l-φ)p, where φis the volumetric fraction of porosity (0 - no porosity and 1 - 100% porosity). At stage 110 and 110A, if Porosity Prediction module 300 is incorporated, the method checks during the iteration, whether a nodal pressure is lower than a porosity forming pressure (PFP). If, so, that pressure is reset to PFP and the incremental gas volume in the same control volume is calculated from the discrepancy of shrinkage volume and net inflow volume. The porosity is updated. PFP is a unique value for each alloy and is defined by the user. The method move on to stage 111 if porosity prediction module is not incorporated, or there is no module pressure that is below PFP. At stage 111, the method checks if the plunger is under pressure control. If yes, the method determines the plunger's new position based on the volume of material squeezed into the die cavity to compensate for the shrinkage. At stage 112, the method proceeds to solve for the temperature fields for both the die and the melt by solving the appropriate energy equations (3). If user has elected to incorporate the Shrinkage Porosity module 300 and the Flow in the Shot Sleeve module 200, the heat transfer between the shot sleeve and the plunger and the melt will also be included in this temperature computation. The update die temperature will be used to determine the heat transfer coefficient between the die and the HTF. The energy equations are preferably discretized according to finite element formulation. Preferably, the energy equation for the melt includes terms for inertia, advection, diffusion, volumetric heat source, viscous heating, and coincident interface heating. Additional, Dirichlet and Neumann boundary conditions should be included as well. The energy equation for the die preferably includes terms for conduction and radiation. After the assembly process, the left- hand side of the global energy equation can be represented by a coefficient matrix operating on a temperature solution vector. The right hand side is made up of a source term vector. This set of discretized form of the energy equations can be solved to yield the temperature at each node. In the illustrated embodiment, the discretized energy equation is solved iteratively. The temperature fields of both the melt and the die is updated. Fraction of solid at each node may be determined using a temperature versus fraction of solid curve. Such curve can be empirically determined, and is generally known in the art. At stage 113, if the user has elected to incorporate the HTF Line in Die module 400 in the simulation, the step sequence of module 400 would have incorporated before solving the energy equation. The flow domain would also include the heat transfer lines in the die. The inlet temperature, flow rate and the type of fluid are defined. Module 400 assumes the ATF lines are circular, and uses established flow and thermal models to calculate the heat flux. The heat flux from the die to the heat transfer fluid, a result of module 400, will be used to predict the die temperature. The sequential steps of the HTF Line module 400 will be discussed later. At stage 115, the method updates all the material properties. At stage 116, the method checks for convergence of the velocity and temperature fields. The method iteratively approximate a solution to the flow equations and energy equations until all velocity and temperature values converge to a user supplied tolerance. If either of the nodal or gauss point velocities or temperatures, I",-, has not converged, the method returns to stage 107 where the mass, momentum and energy equations are formed and iterated again using the updated material properties. At stage 117, the method check for stop filled or stop dwell conditions. The user may define the stop-fill or stop dwell conditions which may be time, fill fraction, or pressure. Fill fraction is preferred as the stop-fill criterion and time is the preferred stop dwell criterion. If the stop time has not been reached, the method returns to incrementing time for another time interval, at stage 106, and proceeds with another iteration. If the stop flow simulation time has been reached, then the mold-filling portion of the simulation is completed. However, the thermal portion of the simulation continues if the user has elected to incorporate the Die Lubricant Cooling module 500 in the simulation. In step 118, the part is rejected and the method continues to determine the heat loss from the die to the spray lubricant. This heat flux will be used a boundary condition to compute the temperature of the die. At the completion of operation sequence of module 500, the simulation is completed. The sequential operations of the Die Lubricant Cooling Module 500 will be discussed later. At stage 119, the method checks for end of cycle conditions. If the number of cycle defined by the user has not been reached, then the updated material properties are applied to the model and a new cycle is started. At stage 120 and stage 121, the user may activate the Mend Line Prediction module 600 to determine the location of mend lines and evaluates the quality of mending of the previous simulation. Mend line prediction is a post-processing activity and does not interact with the computational cycle of the overall simulation sequence 100. The result of the mend line predicts model may be graphically displayed. The sequential operations of the Mend Line Prediction Module 600 will be described later. FLOW IN THE SHOT SLEEVE MODULE The simulation of the flow in shot-sleeve is fully coupled with the simulation for cavity filling and dwell. By incorporating flow in shot sleeve, we can improve the prediction's accuracy due to variations in the initial temperature and heat loss of the feedstock in the shot sleeve, the non-uniform velocity and temperature field at gate, the size and solidification of biscuit and the effects of dwell pressure. Similar to the actual process control, the user can choose to control the ram speed profile or the injection pressure profile. When using speed control, the ram's speed is applied as velocity boundary condition on the adjacent nodes and the ram position in each time interval is monitored. For each control volume adjacent to the ram, the volume of fluid displaced by the ram is determined and incorporated as part of the net inflow in the mass conservation equation. If pressure control is selected, the ram pressure is applied as boundary condition on those adjacent nodes during the first iteration. Then the total volume of fluid flowing away from those adjacent nodes is divided by the ram's sectional area to obtain the ram's effective displacement in that time interval. In the following iterations, this displacement is applied as boundary conditions, by using the same algorithm as in ram-speed control, except that the displacement will be adjusted so that the total ram force from the injected material is equal to that of the hydraulic pressure. For thermal balance, the elements of the ram are moved with the ram itself. Those sleeve elements in contact with the ram are identified through careful bookkeeping so that the heat transfer between them can be included in the energy equation. It should be noted that the ram moves through the sleeve's chamber, many elements of the chamber would become occupied and left behind by the ram. These elements are excluded from the computational domain. Referring now to FIG. 7 which shows the flowchart of an embodiment of the Flow in the Shot Sleeve module 200 of the present invention. While module 200 is depicted in FIG. 7 as a complete, independent routine, it is so depicted for conceptual understanding and demonstrative clarity. It is contemplated and understood that it will be readily apparent to those skilled in the art that the sequential operation of Flow in Shot Sleeve module 200 is interspersed between, and relies upon the operation of the overall simulation sequence to function. Module 200 starts at stage 201 with the inputting of the geometric models of the flow domain which includes: the shot sleeve, all the cores, inserts, cavities, gates, runners, and the ram. The flow domain and the ram are then meshed into a plurality of finite elements or finite difference grids and the matrices representing the meshes/grids are entered. At stage 202, the user may define a set of process parameters and boundary conditions including the initial position and the velocity or pressure profiles of the ram. The velocity and pressure profiles, or the combination of both may be defined as functions of time, position or combination of both. Other parameters may be defined by the user include the initial volume and temperature distribution of the billet. Preferably, the initial position and temperature distribution of the billet is defined within a volume in the shot sleeve and cavities. Finally, the thermal physical properties of the material are also input. At stage 203, the master clock advances by an incremental time Aτ. It should be understood that the time clock is the same as that of stage 106 of overall solution sequence 100, and the incremental time Δr is also the same as the overall solution sequence 100. At each time step, either velocity or pressure, but not both, control the ram motion. Therefore at the beginning of each time step, the method first checks whether the ram's motion is being controlled by velocity or by pressure. If the ram is controlled by velocity, the ram's velocity will be assigned to the nodes in the fluid domain nearest to the ram as a boundary condition. If the ram's motion is being controlled by pressure, the ram's velocity from the previous time step will be used as the initial condition. In the subsequent iteration, the ram's velocity increases if the pushing force due the applied hydraulic pressure is larger than the resisting force from the molten material and vise versa. The average pressure on the molten material can be calculated by dividing the ram's pushing force, which is calculated from the hydraulic pressure and the sectional area of the hydraulic cylinder, by the ram's sectional area. This pressure can be applied as a boundary condition on the nodes in the thermal flow domain nearest the ram. At stage 204, the position of the ram is updated. As the ram always moves in its axial direction, the ram position may be determined using book keeping techniques either from the user defined ram velocity profile or from the fluid's volume displaced by the given ram pressure, and the time increment Aτ. In stage 205, the method identifies the sleeve's elements which have become in direct contact with the ram's elements by comparing the ram updated position and the stored matrix of the shot sleeve. The elements on the slot sleeve surface that became adjacent to the ram are to be used in subsequent heat transfer calculation. The chamber's elements, which have been fully occupied by the ram are excluded from the computation domain, preferably. At stage 206, the method updates the pressure and velocity fields which is the same operation as stage 109 of overall simulation sequence 100. However, because of the incorporation of Flow in the Shot Sleeve module 200, the boundary conditions for the computation of the velocity and pressure fields have been modified. In speed control, the ram's velocity is used as boundary condition on the adjacent nodes. The volume of molten metal displaced by the advancing ram is computed from the geometry of the shot sleeve and the incremental ram positions. This volume of molten metal is equated to the mass flux across the fluxing faces of the control volumes. The mass flux is inserted into the equation for mass conservation to form a Poisson equation for pressure. The pressure equation is solved to obtain the pressure field. In pressure control, the ram force, which is the multiple of the hydraulic pressure and the area of the hydraulic cylinder, is divided by the ram's area and used as pressure boundary condition on the adjacent nodes in the first iteration. After the pressure and velocity are corrected, the total volume of fluid displaced by the ram among the adjacent control volumes is divided to obtain the ram's effective displacement in that time interval. In the following iteration, this displacement is used as ram speed control boundary condition as described above, except that displacement is adjusted after each iteration so that the total pressure force from the injected material is equal to the total ram force generated by the given hydraulic pressure. Using the updated pressure field, the global momentum equations are solved for the velocity field. The momentum equations are formed and solved by the Equal Order method. It should be understood that while the Equal Order method is preferred, other method of forming and solving the mass and momentum equations might be used without deviating from the spirit and scope of the invention. Updating temperature field at stage 207 is the same operation as stage 112 of the overall simulation sequence 100. Again, the incorporation of module 200 modifies the boundary conditions for the computation of the temperature field. The modification occurs in the shot sleeves where elements may become in contact with the moving ram's elements. The heat transfer between these non-aligned elements is accounted for by vigorous book keeping of the interfacial boundary. The method checks for convergence of the velocity and temperature fields at stage 208 (same as stage 116 of overall simulation sequence 100). If the velocity field or temperature field has not converged, the method returns to stage 206 for another iteration using updated pressure, velocity and temperature values. If the velocity and temperature has converged, at stage 209, the method checks whether the criteria for the ram control to switch from speed to dwell have been reached. The user may select the value of the criterion for switching. Preferably, the fill fraction of the mold is selected as the criterion for switching, and more preferably, 99% fill fraction of the mold is selected as the criterion for switching. Alternatively, the user can also choose a time or ram position to switch ram speed control into pressure control. When in pressure dwell at stage 210, the pressure and velocity can be solved with the same algorithm as described above. Since the ram's motion is typically small in dwell stage, the pressure gradient among the nodes adjacent to the ram is small. Therefore, the pressure calculated from the ram force and ram's sectional area can be used as boundary condition for the entire iterations. After the pressure field is updated, the motion of the ram is calculated by dividing the total volume of fluid pushed out from those adjacent control volumes adjacent to the ram by the ram's sectional area. At stage 211, the method checks whether end time for dwell has been reached. The end time for dwell is selected by the user. If end time for dwell has not been reached, the method returns to stage 203 and advances a time increment Δτ, and the pressure and velocity fields are updated. At dwell, the driving force for melt flow is pressure rather mass flux. At stage 212, after checking that end time for dwell has been reached, the method continues to stage 118 of overall solution sequence 100. SHRINKAGE POROSITY PREDICTION MODULE The Shrinkage Porosity Prediction module 300 uses a coupled thermal-flow model to predict the formation of shrinkage porosity. The illustrated embodiment is applicable to a semi-solid forming, or other casting processes. First, the thermal- flow model provides a prediction of temperature, pressure and velocities within the region occupied by the metal and the die. Under the constraint of the conservation of mass, a region where alloy is becoming denser over time will results in a low pressure values in the momentum equation if the flow resistance in the surrounding matrix is large enough to retard material from filling in. As the local pressure becomes lower than the porosity-formation pressure, which is typically very low or close to zero, any further shrinkage will then produce porosity in that region. Therefore, the formation and advection of shrinkage porosity can be calculated directly by incorporating porosity and density variation in the mass conservation equation and momentum equation. This process can be undertaken at each time level to produce a time history of porosity at each position in the model. Referring now to FIG. 8 which shows the flowchart of an embodiment of the Shrinkage Porosity Prediction module 300 of the present invention. At stage 301, the user selects the Shrinkage Porosity Prediction module 300. The method incorporates the step sequence of module 300 into the overall flow simulation module 100. The user, in addition to the ram's velocity/pressure during cavity filling, may define the pressure profile as a function of time during the dwell stage. The momentum equations are discretized and expressions for the velocity components are formed as described in stage 109 of overall simulation sequence 100. The effects of porosity on mass conservation is incorporated by defining an apparent density, β = (l-φ)p , where φ is the volume fraction of porosity and p is the theoretical density of the alloy at a given temperature. The continuity constraint is modified to include the effects of porosity. The modified continuity constraint is expressed as: dx dy dz dτ where (21) β= (l-φ)p the apparent density of the alloy x, y, z = three components of the coordinate system r = time u, v, w = three velocity vector components φ = volumetric fraction of porosity (0 - no porosity and 1 - 100% porosity) p = theoretical density of the alloy as function of temperature In the illustrated embodiment, the method follows the Equal Order method to solve the discretized momentum and mass equations through the use of a Gauss point velocity field. The expression for the Gaussian point velocity vector is inserted into the modified continuity constraint to form the Poisson equation for pressure. This equation is solved to determine the pressure at each node within the computational domain. While the Equal Order method is preferred, other methods of deriving and solving the coupled mass and momentum equation may also be used without deviating from the scope of the invention. Analysis for shrinkage porosity commences after the start of pressure dwell. At stage 302, the pressure field is analyzed. Shrinkage porosity forms or increases when the pressure of a control volume drops below a reference pressure, called porosity formation pressure (PFP) which is a unique value for each alloy and defined by the user as part of the input data. The PFP may be defined as the nucleation pressure of gases dissolved in the alloy at the corresponding temperature. Alternatively, PFP may be defined as atmospheric pressure. At this stage, those control volumes with pressure below PFP are identified. If a region containing partial liquid is isolated from the pressurization, for example, i.e. the material surrounding the region has completely solidified, the corresponding shrinkage of the remaining liquid fraction is calculated and added to the porosity. This region is then excluded from the computational domain. The material can be considered as solid and does not flow when its temperature is lower than a pre-determined temperature, e.g. the solidus temperature, defined by the user. This approach is preferred in the present invention to avoid a very large (infinite) viscosity of the material which may exceed the computer's calculation limit. At stage 303, the method resets the pressures of those control volumes having pressures below PFP to PFP, resulting in a reduction of pressure gradients over some of the elements. At stage 304, using the reduced pressure gradients, the method determines the nodal velocities and updates the velocity field. At stage 305, the method determines the volumetric net inflow rate, , in each control volume with pressure below PFP using the reduced pressure gradient. The reduced pressure gradient yields a smaller net flow rate than when the pressure in the specific control volume was lower. The smaller net flow rate may be rationalized as accounting for the space taken up by the voids. At stage 306, the increment in porosity within a control volume is computed by taking the difference between the densification rate and the smaller net flow. The following equation may be used to estimate the incremental volume fraction of porosity, dφ, within the control volume: (22) where p is the theoretical density at a given temperature V is the volume of the control volume v is the net flow rate of mass coming into the control volume dτ is the incremental time step When the dwell stops (pressure at the gate or at the ram is relieved), the method checks for the presence of any non-zero liquid fraction remains in the flow domain. If the non-zero liquid fraction remains, the method calculates the corresponding shrinkage and updates the final porosity. At stage 307, the apparent density β, at each node is calculated using the updated porosity value φ. The updated apparent density ?is used in computing the velocity and pressure fields in the next time step. A time history of shrinkage porosity at each control may therefore be established. At stage 308, if the Flow in the Shot Sleeve module 200 is also selected in the same overall simulation sequence 100, the method first updates the ram's new position based on the volume of material squeezed into the cavity to compensate the shrinkage, and continues to stage 109 of overall simulation sequence 100. HEAT TRANSFER FLUID LINE IN DIE MODEL The HTF Line in Die module 400 models the heat transfers between the die surface and the heat transfer fluid, such that the temperature field may be determined accurately and processing conditions may be set properly. Modeling heat transfers between two 3-dimensional objects are complex and a solution is not always attainable. This model assumes that the HTF lines have approximately circular cross sections, such that the axis-symmetric velocity and temperature distribution of the HTF can be calculated based on well-established analytic solutions. With simple accounting technique, the interface, where the heat transfers occurs, between the die surface and the HTF line may by accurately determined. This model, thus, simplified a three-dimensional problem to a one- dimensional problem in which the solution is manageable. As the HTF lines are typically drilled with circular cross section, this model provides realistic boundary conditions for the energy balance of the flow domain. FIG. 9 shows a schematic diagram of the heat transfer line 60 in relationship to the die elements 61. The entire heat transfer line 60 is divided into a number of linear segments kj, wherey=l,2,...N. Each segment k j is associated with a known set of die element surfaces 62 at where the heat transfer from the die to the cooling channel is accounted. Each segment k j is bounded by an upstream node lj.j and a downstream node lj, named in relation to the flow direction m. The average temperature distribution within each segment k j is linearly interpolated from the nodal temperature at its two ends. The governing equation for the this heat transfer is an implicit form of a one- dimensional energy equation: where A fl o W = the cross sectional area of the HTF line L = length of the flow segment Cp = specific heat P = density of the HTF Δτ = time step h k = convection heat transfer coefficient between the HTF and the die segment k A = circular area of the HTF line in segment k T mk = mean die temperature in segment k T j = temperature of the HTF at node j at the end of a time step T°j = temperature of the HTF at nodej at the beginning of a time step By defining the following terms, Δ τ / = A flow L p C p (24) (25) N f E = ∑ h k A k , (26) an update of the cooling channel temperature, T j , can be expressed as, where, (27) (28) c = 2IEτ + T 0 j + T (29) (30) The method of the present invention solves the above energy equation numerically using a tri-diagonal solver. At each time step, the Reynolds number (Re), Prandtl number (Pr), Nusselt number (Nu), and the corresponding heat transfer coefficient h on each element surfaces of the heat transfer line is calculated based on the user defined volume flow rate, the thermal physical properties of the HTF, and the updated HTF temperature. The heat flux exchanged between the HTF and the die is calculated on each die element surface 62 from the heat transfer coefficient h k , the local temperature difference between the HTF and the die, the element surface area and the time step. With the heat flux of the die element surfaces 62 within each line segment k j , the HTF's temperature at the upstream node I j .j, the volume flow rate and thermal physical properties of the HTF, the HTF's thermal energy, and setting the boundary condition that the temperature of the first node To = T in ι et , temperature at the downstream node T } can be calculated. By this approach, the analytical/empirical correlation between Re, Pr and Nu and the variation of the HTF's flow rate, temperature and viscosity can be incorporated to predict the heat transfer accurately without heavy increase in the computation steps. Referring now to FIG. 10 which shows the flowchart of an embodiment of the Heat Transfer Line in Die module 400 of the present invention. At stage 401, the user selects to include module 400 in the overall simulation sequence 100. The user may define the dimension and locations of the HTF lines 60 in the finite-element model of the die, the type of fluid, its inlet temperature, volumetric flow rate and initial temperature. The method identifies all of the die element surfaces 62 associated with HTF line 60 from the finite-element mesh of the die and user specified element edges of the inlet and outlet of the HTF lines. Then, these HTF element surfaces are grouped sequentially, based on their flow distance from the inlet, into a number of segments k j where j = 1, N. The number of segments, N, is determined by the user based on the total length of the HTF line. At stage 403, the method updates the viscosity, μ, of the HTF based on the temperature T k of the heat transfer fluid at the beginning of a time step. Initially, T k is a boundary value provided by either the user or the method. As the flow progress, T k is updated from the accumulated heat which the HTF absorbed from the die. Based on the temperature of the HTF, the method determines the density, the viscosity, μ, and the thermal conductivity of the HTF at segment kj, . These parameters may be directly read from look-up data tables which correlates these parameters with temperatures for common HTFs. The data tables are commonly available and are known to those skilled in the art. The Reynolds number, Re = pvdlμ, and Prandtl number, Pr = C p μ/k, axe calculated from the known properties of the HTF and the diameter of the HTF lines. The corresponding Nusselt number, Nu, can then be determined from the Reynold and the Prandtl numbers. The Nusselt number may be determined from the Re and Pr numbers based on empirical correlation. Alternatively, the Nusselt number may be determined based on an analytical equation. The relationship between Re, Pr and Nu numbers are model dependent. It is known to those skilled in the art which particular model best represents their processes. At stage 404, the method determines the heat transfer coefficient h k in segment kj from the Nusselt number according to the following relationship: h k = Nuk/d where hk = heat transfer coefficient of the HTF in segment k k = thermal conductivity d = diameter of the HTF line At stage 405, the method inputs T mk , the average nodal temperatures of the set of die elements 61 associated with segment k of the HTF line into Equation (23). The nodal temperatures of the die elements 61 are read from the die temperature field of a previous iteration. The heat flux between the HTF and the die may then be determined on each die element surface 62 from the heat transfer coefficient, hk, the local heat and temperature difference between the die and the HTF, the die element surface area and time. The method further calculates the total rate of thermal energy absorbed by the HTF in a HTF line by integrating all of the heat flux on it interface with the die. The rate of thermal energy absorbed is saved at each time step and be plotted as a function of time when request. At stage 406, the method builds a transport equation and forms a tri- diagonal matrix to simultaneously solve for all the nodal temperatures of the HTF. The average thermal energy of the HTF at the down-stream node of a segment is calculated by adding the thermal energy of incoming HTF and the energy absorbed by the HTF from the die and dividing the total by the mass flow rate of the HTF. Given the boundary condition that at T j= o = T m n, the HTF's average temperature at each down-stream nodal point may then be determined sequentially, segment by segment, from its average thermal energy and thermal properties, i.e. specific heat. At stage 407, the updated nodal temperatures of the HTF are then used as boundary conditions to calculate the heat flux between the die and the HTF. The newly calculated heat flux is incorporated in the die's energy equation as thermal boundary condition to solve the die's new temperature. DIE LUBRICANT COOLING MODULE The Die Lubricant Cooling module 500 calculates the amount of heat removed from the die by the spraying of lubricant after the part was ejected. This heat flux is incorporated in the die's energy equation as thermal boundary condition to solve the die's new temperature. In this module, the element surfaces covered by the lubricant spray are identified mathematically by emanating a spray cone from the nozzle's position toward the mold surface. By comparing the relative position of the element surfaces on the mold cavity with the spray cones, those element surfaces covered by the spray can be identified. If the spray nozzle moves, the position of the spray cone and the corresponding sprayed element surface will be updated for every time step. The cooling coefficient on an element surface is determined empirically. In actual set up, the user only needs to define the location, angle and motion path of each spray nozzle, nozzle type, spray pressure and flow rate. The program will automatically identify and assign a time- dependent convection-cooling coefficient at every time step on those element surfaces intercepted by the spray cone. With this function, the user can quickly change the spray nozzle, spray pattern and spray parameters to evaluate their impact on die temperature and product's quality. Referring now to FIG. 11 which shows the flowchart of an embodiment of the Die Lubricant Cooling module 500 of the present invention. Module 500 commences after the stop flow time for the flow portion of the simulation has been reached, the die opens and the part is ejected. If a user selects to incorporates module 500 to the overall simulation, the user may define the spray parameters at stage 501. For example, the user may specify the style of the spray nozzle, e.g. the shape and size of the nozzle, its initial location relative to the die surface and its motion profile. The nozzle can be of any shapes, but preferably, it is either circular or elliptical. In addition, the user may also specify the spray parameters, such as the lubricating fluid, its thermal properties, flow rate, the spray pressure, and spray time. The spray nozzle's motion is defined either by its initial position and orientation vectors and subsequent displacement and rotation vectors, or by successive position and orientation vectors, at variable time intervals. In either case, the vectors can be defined relative to the global coordinate of the entire finite- element model or a local coordinate defined on the die. If a local coordination system is used, preferably, the nozzle is positioned at the origin at the origin of this coordinate system, such that the major and minor axis of the nozzle form the x and y axes and the line extending from the nozzle to the die face forms the z-axis. The finite element model of the die is then transformed into the same local coordinate system for consistency. While the above described coordinate system for modeling the spray cone is described, other coordinate systems may be used. The Die Lubricant Cooling module 500 is integrated into the global energy balance of the thermal flow simulation, its operation sequence is controlled by the master clock of overall simulation sequence 100. At stage 502, when the master clock advances a time interval AT, a new time step is also started for module 500. At the beginning of each time step, at stage 503, the position of each lube spray nozzle is updated based on the given motion profile. At stage 504, a mathematical representation of the nozzle's 3D spray cone may be calculated from the nozzle's shape and its orientation. At stage 505, the elements inside the spray cone are identified and those elements which are hidden behind other surfaces are excluded. The above identification and exclusion is accomplished by a visual test, which the nozzle is placed at the eye position and viewed along the Z-axis of the local coordinate system towards the die. The die element faces which are in front of the spray and have a normal vector towards the spray are selected. Upon the element surfaces that are selected, a second test is performed to determine which element faces are actually exposed to the spray (i.e., within the cone of influence), and not being hidden behind other die features. The test is: x; if + < z, a (31) then, the element face is determined to be inside the cone of influence. If not, the element face is excluded from the subsequent modeling. The subscript, I, denote the local coordinate system as defined by the spray. At stage 506, the view space (projected area) of each of the die element faces within the spray cone are determined by a hemispherical projection. Hemispherical projection techniques are well-documented in radiation heat transfer literature and are well known to those in the art. For example, Heat Transfer - A Basic Approach, by M.N. Ozisik, published by McGraw Hill, 1985 describes a hemispherical projection technique. At stage 507, the heat flux from each of the die element faces within the cone of influence is computed. This heat flux is incorporated in the die's energy equation as thermal boundary condition to solve for the die's new temperature at the next step. The heat flux or loss from the die cavity surface can be determined from the projected areas of the die element faces, the temperature of the die, the temperature of the lube and the ambient temperature and the spray parameters. Generally, the heat fluxes are computed using heat transfer coefficients which are functions of the die temperature and the spray parameters, such as spray volume, spray flow rate, spray pressure etc. Preferably, the heat transfer coefficients are predetermined and stored as look-up tables. Tables of heat transfer coefficient may be constructed empirically. The following is an example of the construction of such a table. First the volume of spray reaching the projected area of the element face is computed by the expression: Q f = Q s r? s f hp (32) Q f = volume of spray reaching mold face, Q s = Volumetric spray flow Where, η s = Spray carry out efficiency f hp = fraction of the view space Then spray experiments can be conducted where a known volume of lubricant is sprayed on a heated flat plate. All thermal physical parameters of the flat plates are known. Preferably, the plate is of similar material as the die. The temperature drop resulted from the spray is measured, and the heat transfer coefficient computed. The volume of spray used versus heat transfer coefficients are stored. The table can then be used to look up heat transfer coefficient when the volume of spray reaching the element faces of the die is known. While a suggestion is made as how a look-up table for heat transfer coefficient is established, other methods, empirical or numerical, of determining a heat transfer coefficient from the projected area of a die element face may also be used. After the heat flux is calculated and the die temperature is updated, at stage 508, the method checks whether a user defined stop spray time has been reached. If the stop spray time has not been reached, the sequence returns to stage 502 of incrementing time by an interval and repeats the spraying sequence. If the stop spray time has been reached, the Die Lubricant Cooling module 500 is completed; and the simulation continues to stage 119 of overall solution sequence 100. MEND LINE PREDICTION MODULE The Mend Line Prediction module 600 simulates the location of the mend line in a fluid flow. Module 600 predicts the position of the mend line from the flow and thermal data collected during a mold filling simulation. The module is executed after the completion of the simulation. It is a post-processing activity and does not interact with the computational cycle of the overall simulation sequence 100. However, it is contemplated that the mend line may be tracked as the flow simulation progresses. Along the free surface of the melt front, mathematical points are created to represent material particles. Visually, these particles are represented by a combination of colors and symbols. The position of each particle is tracked. At each time step, particles are advected to a new position based on the calculated melt velocity on the previous position. After the simulation is completed, the user can display the distribution of these particles which were assigned different colors or symbols denoting the time of birth (when they were added) or the local temperature. In this way, meld lines can be identified. Referring now to FIG. 12 which shows the flowchart of an embodiment of the Mend Line Prediction module 600 of the present invention. When the Mend Line Prediction module 600 is activated, the user may define the target particles density, e.g. number of particles per unit area. On the initial position of the melt front, particles are created on the surface based on the target particle density chosen. Further, the user may define the start time and end time for the simulation to predict the formation of mend lines only at a specific time period within the overall simulation. At the start of the simulation, stage 601, the time clock is initialized. Being a post-processor simulation, the time step of module 600 is not governed by the master clock of the overall simulation of the thermal flow. At stage 602, the time clock advanced a time interval Δτ. The time interval may be defined by the user. The user may follow the same time interval Δrused during the simulation when the flow data was collected. Alternatively, the user may define a time increment according to the desired resolution for the motion of the melt. At stage 603, the position and shape of the melt front in each partially-filled control volume is mathematically identified based on the volume fraction of the melt, j„, which is defined as: (35) where V m is the volume of the control volume occupied by the melt V is the total volume of the control volume In one embodiment of the present invention, the free surface of a melt front is defined as the surface where the volume fraction of the melt, as interpolated from the nodal points, is 0.5, i.e., f m = 0.5. The elements where the melt front exists are first identified from the volume fraction of their nodes. In an element containing a melt front, the melt volume fraction would be larger than 0.5 for some nodes and less than 0.5 for other. The point where the volume fraction of the melt equals to 0.5, intersects with the edges of the element is then computed. The resulting collection of points is used to define a polygon whose centroid is denoted by the free surface position vector Xf S .. While melt volume fraction is used to define a melt front, many other definitions may be used without deviating from the scope of the invention At the start of the simulation, the method creates material particles along the free surface of the melt front. The particles are mathematical points represented visually by an indicia. The indicia may be a color or symbol or a combination thereof. Preferably, particles that are created at the same time and at the same melt front are assigned the same indicia. Initially, each particle are also assigned a position vector, X p °, a temperature T p ° and a velocity vector V p °. The initial velocity and temperature of the particles are equal to the velocity and temperature of the associated melt front, as interpolated inside an element from the nodes. The position, temperature and velocity of each particle are tracked. As the fluid flow progresses, the melt front may expand or contract, causing a change in the particle density along the meld front. At stage 604, the method determines whether the number of particles at the melt front is sufficient. The method counts the number of particles which are considered attached to the melt front. The method compares the position vector of each particle, X p to the average position Xf S of the free surface . If the particle is within a user-defined tolerance of the free surface points, then the particle is defined as an attached particle. If the particle is outside the defined tolerance of the free surface points, then it becomes a detached particle. In this illustrated embodiment, the method uses the distance between a particles and the melt front as the judging criterion. If the distance between a particle from the free surface of the melt front is within a user-defined value, the particle is considered attached, i.e. on the melt front. While a distance criterion is used to determine whether a particle is still attached to the melt front, other criteria, such as a range of fraction of liquid, particle density, etc., may be used without deviating from spirit and scope of the invention. If the number of particles on a melt front is lower than the user-specified number density, the module adds new particles to the free surface at stage 605. The method assigns different color or symbols to track these newly added particles. The time, when each particle is created is also recorded. Again, once created, the position, temperature and velocity of each particle are stored and tracked continuously. If the number of particle at the free surface is judged sufficient, the module continues to stage 606. At stage 606, the positions of all particles are updated. At each time step, the particle is adverted to a new position based on the calculated melt velocity on the previous position. The new position is updated using the local velocity vectors: X i+1 = X i + V-Aτ (36) where X is the particle position vector Vis the fluid's velocity vector at the particle's position Δris the time step i, i+1 axe time steps During these advection processes, the particles are constrained such that they can not pierce the boundaries of the model. The new position is stored. At stage 607, the average position of the free surface within each element is updated. At each time step, the volume fraction of the melt in each control volume changes because of the incremental net inflow of the melt, the position of the melt front will have to be adjusted accordingly. The position of the melt front is updated at each time step. At stage 608, the method updates the attach/detach status of each particle following the procedure previously described. If a particle no longer attached to the melt front, the symbols which was originally assigned to the now detached particle is changed to a different color/symbol for ease of visual identification. At stage 609, the positions of the particles are rendered to the computer's screen for visual analysis. The attached and detached particles are distinguished by color/symbols. Visually, at where particles of different symbols and colors, which represent different melt fronts, collide is the location of the mend lines. Further, the quality and strength of mending may also be evaluated visually by the temperature and the level of mixing of the particles. The higher the temperature and the thoroughness of the mixing of the particles at the mend line, the better the mending would be. At stage 610, the method checks for the stop time. If stop time has not been reach, the method returns to stage 602, and repeat the simulation at a different time step. If stop time has been reached, the simulation is completed. User may output result of the simulation. While the invention has been illustrated and described in detail in the drawings and foregoing description, the same is to be considered as illustrative and not restrictive in character, it being understood that only the preferred embodiment has been shown and described and that all changes and modifications that come within the spirit of the invention are desired to be protected.

Claims: What is claimed is: 1. A method for simulation thermal flows for casting/molding processes, comprising: (a) providing at least one modules for incorporation in a simulation, wherein said modules are selected from a group consisting of a Flow in Shot Sleeve module, a Shrinkage Porosity Prediction module, a Heat Transfer Fluid Line in Die module, and a Die Lubricant Cooling module; (b) discretizing a flow domain into finite elements; (c) updating mass balance in said flow domain; (d) updating velocity and pressure fields; (e) updating temperature field. 2. A method for simulating a thermal flow in a casting/molding process comprising: selecting at least one component of a casting/molding process to simulate, wherein said component is selected from a group consisting of thermal flow in a shot sleeve, formation of shrinkage porosity, heat flux to heat transfer fluid, cooling effect of die lubricant, and location of mend lines; discretizing a computational domain into a mesh of elements; updating material balance in said computational domain; updating pressure and velocity field; updating temperature field; and repeating said updating of material balance, pressure, velocity and temperature until a stop simulation criterion is reached. 3 The method of claim 2, wherein said determining pressure and velocity fields comprises deriving a discretized momentum and a mass equation according to a finite element formulation, and solving said equations following an Equal-Order method. 4. The method of claim 3, wherein said finite element formulation is a control volume based finite element formulation. 5. The method of claim 2, wherein said thermal flow in shot sleeve component is being simulated, wherein said velocity and pressure is determined from velocity of a ram in said shot sleeve when said ram is under velocity control, and wherein said velocity and pressure is determined from a balance of ram force and resisting force from said fluid when said ram is under pressure control. 6 The method claim 2, wherein said determining the temperature field of said computational domain further comprises determining cooling effect of lubricant sprays, and wherein said cooling effect of lubricant sprays is determined by automatically identifying an empirically determined cooling coefficient and calculating heat loss from said cooling coefficient. 7. A method to incorporate heat transfer and fluid flow in a shot sleeve in a casting simulation comprising: incorporating said shot sleeve and ram into a flow domain, wherein motion of said ram is controlled either by velocity or by pressure; determining mass flux across elements adjacent to said ram, wherein said mass flux determination is based on velocity of said ram when said ram is under velocity control, and wherein said mass flux determination is based on a balance between a pushing force of said ram and a resisting force of said fluid when said ram is under pressure control from said; updating pressure and velocity; updating temperature field by assuming non-coincident heat transfer between said ram, said shot sleeve and said flow. 8 The method of claim 7, wherein said pushing force of said ram is calculated from the hydraulic pressure applied and sectional area of a hydraulic cylinder. 9. The method of claim 8, wherein said updating said pressure further comprises calculating a pressure boundary condition for nodes of element adjacent said ram by dividing said ram force by ram's surface area. 10. The method of claim 9, wherein said updating said pressure further comprises using said pressure boundary condition for a first iteration, calculating fluid displacement, and using said displacement of as ram velocity boundary condition, and repeating iteration until said pushing force of said ram equals said resisting force of said fluid. 11. The method of claim 7, wherein said non-coincident heat transfer is accounted for by book keeping method. 12 The method of claim 11, wherein said updating temperature comprises identifying a first set of elements of said shot sleeve which became adjacent to said ram and including said first set of element into heat transfer computation, and wherein said updating temperature further comprises identifying a second set of elements of said shot sleeve which became fully occupied by said ram and excluding said second set of elements from said computational domain. 13 The method of claim 12, wherein said first and second sets of elements are identified by a book keeping technique. 14. The method of claim 7, further comprising providing a velocity profile, said dwell pressure profile for said ram, said velocity profile and said profile are provided by a user. 15. The method of claim 14, wherein switching from said velocity controlled of said ram to pressure controlled is determined by a start dwell condition. 16 The method of claim 15, wherein said start dwell condition is a fill fraction of the mode and wherein said fill fraction is at 99% fill. 17. The method of claim 15, wherein said start dwell condition is time. 18. The method of claim 15, wherein said start dwell condition is position of said ram. 19. A method to incorporate porosity into a melt casting-mold filling simulation comprising: incorporating effects of porosity into continuity constraint; starting pressure dwell, identifying elements having pressure below porosity formation pressure, determining velocity and net melt inflow and incremental volume fraction of porosity based on said reduced pressure gradient; updating porosity. 20 The method of claim 19, wherein said modifying comprises incorporating an apparent density β, into a continuity constrain such that said modified continuity constraint is expressed as: dfli ^ ^βy { dβ = Q where β = (l-φ)p apparent density of the alloy x, y, z = three components of the coordinate system τ = time u, v, w = three velocity components φ = volumetric fraction of porosity (0 - no porosity and 1 - 100% porosity) p = theoretical density of the alloy as function of temperature 22 The method of claim 21, said porosity forming pressure is a pressure at which gases dissolved in said melt at a corresponding temperature will nucleate. 23 The method of claim 21 , said porosity forming pressure is atmospheric pressure. 24 The method of claim 23, wherein said updating comprises computing an incremental porosity by an expression as follows: where p s is the theoretical density at solidus temperature p is the theoretical density at a given temperature V is the volume of the control volume V is the net flow rate of material coming into the control volume dτ is the incremental time step 25. The method of 24, wherein said updating of porosity further comprises identifying elements which contain liquid melt but is surrounded by elements containing melt having temperatures below a pre-determined temperature, calculating porosity resulting from solidification of said liquid melt and adding said calculated porosity to total porosity. 26 The method of claim 25, wherein said pre-determined temperature is a temperature at which said melt solidifies. 27 claim 26, wherein said updating or porosity further comprises checking, after said dwell stage, for non-zero liquid melt fraction in said mold, calculating porosity resulting from solidification of said non-zero liquid melt and adding said calculated porosity to said total porosity. 28. The method of claim 27, wherein said updating porosity further comprises computing a net inflow rate for each of said control volumes which had said negative pressure. 29. The method of claim 28, further comprising the step of advancing time by an incremental time and computing velocity and pressure fields using said updated apparent density, β, thereby establishing a time history of said shrinkage porosity at each said control volumes. 30. A method which incorporate heat transfer between a die and the heat transfer fluid (HTF) into a thermal flow comprising: modeling a HTF line as having approximately circular cross sectional geometry, wherein well-established analytical solution for velocity and temperature and analytical/empirical correlation between Re, Pr and Nu numbers exist; modeling heat transfer between said die and said HTF as a one-dimensional problem; determining heat transfer coefficient of said HTF; updating temperature. 31 The method of claim 30, wherein said modeling comprises dividing said HTF line into a plurality of segments, each said segment is bounded by a upstream node and a downstream node, wherein temperature of said HTF is evaluated at said nodes and temperature distribution of said HTF within each segment can be linearly interpolated from two said nodes; associating each said HTF line segments with a set of die element surfaces. 32 The method of claim 31 , wherein said modeling further comprises defining a one-dimensional energy equation which governs heat transfer between said HTF and said die as follows: where A f i ow = the cross sectional area of the HTF line L = length of the flow segment C p = specific heat p = density of HTF Δτ = time step h k = convection heat transfer coefficient between the HTF and the die in segment k Aj c = circular area of the HTF line in segment k T mk = mean die temperature in segment k T j = temperature of the HTF at node j at the end of a time step T° j = temperature of the HTF at node j at the beginning of a time step 33. The method of claim 32, wherein said density, viscosity and thermal conductivity of the HTF is determined from temperature of said HTF. 34. The method of claim 32, wherein said viscosity, density and thermal conductivity of said HTF is determined from look-up tables. 35. claim 33, wherein said heat transfer coefficient is determined from HTF's Nusselt number which in turn is determined from said HTF's Pandtl and Reynold's numbers, and said Pandtl and Reynold's numbers is determined from said viscosity and diameter of said HTF lines. 36 claim 35, wherein said Nusselt number is determined from said Re and Pr numbers based on an analytical equation. 37 claim 35, wherein said Nusselt number is determined from said Re and Pr numbers based on empirical correlation. 38 claim 35, wherein said heat flux is determined from surface area of said segment, heat coefficient and difference of average temperature of said die elements and said HTF. 39 claim 35, wherein temperatures at said nodes of said HTF lines are determined simultaneously by solving said energy equation using a tri-diagonal matrix comprising: (a) defining terms of said transport equation as: Δ τ A flow L p C p (b) expressing the cooling channel temperature as: where, b = 1 + I (E 2m C P ) c = 2IEτ + T 0 j + T °j - ι 40 The method of claim 39 further comprising determining total rate of thermal energy absorbed by said HTF by integrating all of said heat transfer over said set of die element surface and plotting said rate of energy absorbed as function of time. 41. The method of claim 30, further comprising updating said temperature field of said overall thermal fluid flow simulation. 42. A method to incorporates cooling effect of lubricate spray on a die into a melt casting simulation comprising: providing process parameters comprising spray parameter and motion profile of a spray nozzle; updating position of said nozzle based on said motion profile; determining a spray cone of said nozzle and die element surfaces covered by said spray cone; identifying automatically a cooling coefficient based on a built-in model; determining heat lost from said die surface from said cooling coefficient; and incorporating said heat lost into said die's energy equation to determine temperature field of said die. 43 claim 42, wherein said motion of said spray is defined in a local coordinate system, and a finite element mesh of said die is transformed in said local coordinate. The method of claim 6a, wherein said local spray coordinate system of further comprises a transformation matrix which places elements of a three-dimensional mesh that makes up said die into said local spray coordinate. 44. The method of claim 43, wherein said local spray coordinate system comprises an origin, and x-, y- and z-axis, and wherein said origin is defined at said spray nozzle, said x-axis is defined by a major axis of said spray nozzle, said y-axis is defined by a minor axis of said spray nozzle, and said z-axis is defined by an axis extending from said nozzle to said die. 45. The method of claim 44, wherein said motion of said spray nozzle is defined by its initial position and an orientation vector and subsequent displacement and rotation vectors. 46 The method of claim 45, wherein said motion of said spray nozzle is defined by successive position and orientation vectors at various time intervals its initial position and an orientation vector and subsequent displacement and rotation vectors. 47 The method of claim 46, wherein said spray cone is determined from a geometric analysis of shape of nozzle, and orientation. 48 The method of claim 42 wherein said determining of die surfaces covered by spray cone comprises placing said nozzle at an eye position and viewing along said z-axis of said local coordinate system toward said die and identifying die element faces which are in front of said spray and having a normal vector facing said spray, excluding elements that are hidden behind other surfaces. 49. The method of claim 48 wherein said excluding of said elements comprises performing a zone of influence test. x, if + yι < z, a 50 The method of claim 49, wherein said cooling coefficient is determined from a look-up table. 51 The method of 50, wherein said look-up table is empirically determined following a procedure comprising: (a) computing a volume of spray reaching said projected area of said die element face by the expression: Q f = Qs fh P Q f - volume of spray reaching mold face, Q s = volumetric spray flow Where, η s = spray carry out efficiency f hp = fraction of the view space (b) spraying a known volume of said lubricant on a heated flat plate, wherein all thermal physical parameters of said heated flat plate are known; (c) measuring temperature drop resulting from spraying of said known volume of lubricant; (d) computing a cooling coefficient corresponding to said known volume of lubricant spray used; (e) storing said cooling coefficient relating to said volume of spray used in a look-up table. 52. A method to predict location of mend lines of a melt flow comprising: providing a processor adapted to display graphics; creating particles at melt fronts, tracking said particles as said melt flows; displaying position of particles as a function of time, identifying visually position of said mend lines as where particles having different symbols collide. 53. The method of claim 53, wherein said particles are represented with symbols, wherein said symbols are different for particles created at different time and at different melt front; 54 The method claim 52 , further comprising providing data from a thermal flow simulation 55 The method claim 53, further comprising evaluating quality and strength of mending from level of mixing of particles represented by different symbols. 56 The method of claim 55, wherein said stop condition is user determined. 57. The method of claim 55, wherein said incremental time is user determined. 58. The method of claim 57, wherein said incremental time is same as an incremental times used in said thermal flow simulation. 59. The method of claim 52, wherein said free surface is defined from a volume fraction of said melt. 60 The method of claim 52, wherein updating position of said particle comprises a procedure of calculating new positions of particles from a mathematical expression as follows: 61 The method of claim 52, further comprising determining sufficiency of said particles at said melt front. 62. The method of claim 61, wherein determining sufficiency of said particles comprising determining number of particles that are attached to said melt front. 63 The method of 62, wherein a particle is considered attached to a melt front if a distance between said particle and melt front is within a predetermined tolerance. 64. The method of claim 52, wherein said updating position of said material particle comprises the step of advecting each said particles to a new position and said new position is determined by a local velocity vector of said melt.

Inventors: NIEH JENN-YEU, WAITE DAVID MARC, WANG SHAUPOH, NIEH, JENN-YEU, WAITE, DAVID, MARC, WANG, SHAUPOH
Applicants: AEMP CORP, NIEH JENN YEU, WAITE DAVID MARC, WANG SHAUPOH, AEMP CORPORATION, NIEH, JENN-YEU, WAITE, DAVID, MARC, WANG, SHAUPOH
Application year: 2001
Main classification: G06G7
IPC codes: B22D17, B22D46, B29C45, G06F17