The invention relates to the field of three dimensional modeling of fluid flow in a cavity and, more specifically in one embodiment, to the modeling of an injection molding process for producing molded polymer components. 
The use of injection molded plastic components has dramatically increased in many industries in recent years.  Manufacturers of electronic equipment, consumer goods, medical equipment, and automotive parts are producing more and more of their products and components used in their products out of plastics than ever before.  At the same time, competitive pressures are driving manufacturers in the plastics injection molding industry to find new methods to optimize the designs in order to better match the designs to the production process.  When the need for component or mold configuration modifications are discovered late in the design development process, the delay and associated costs to implement the necessary changes rise rapidly.  Companies that want to ensure that their components are producible and will perform optimally have begun to use computer aided engineering techniques to simulate or model the complex flows in an injection mold, in order to understand better the manufacturing process and integrate this knowledge into component design, early in the design phase. 
There are a number of factors that should be considered when designing an injection mold and the component which is to be produced therein.  Parameters such as the overall component geometry, minimum and maximum wall thicknesses, number and location of gates in the mold through which the liquid polymer is injected, number and location of vents in the mold through which gas in the cavity escapes, polymer composition and properties, and shrinkage allowances, are a few.  Due to the closely interrelated relationship, component and mold design cannot reliably be based purely on form and function of the end component, but should also consider the effects of the manufacturing process. 
Computer aided engineering simulation can be used advantageously to provide design and manufacturing engineers with visual and numerical feedback as to what is likely to happen inside the mold cavity during the injection molding process, allowing them to better understand and predict the behavior of contemplated component designs so that the traditional, costly trial and error approach to manufacturing can be eliminated substantially.  The use of computer aided engineering simulation facilitates optimizing component designs, mold designs, and manufacturing processing parameters during the design phase, where necessary changes can be implemented readily, with the least cost and impact on schedule. 
A basic discussion of the injection molding process and the challenges associated with producing high yield, quality injection molded components is addressed in the primer xe2x80x9cMoldflow Design Principles: Quality and Productivity by Design,xe2x80x9d distributed by Moldflow Pty.  Ltd., Kilsyth, Victoria, Australia, the assignee of the instant patent application, the disclosure of which is herein incorporated by reference in its entirety. 
Briefly, the injection molding process is a complex, two step process.  In the first step, referred to as the filling phase, polymer material is forced under pressure into the mold cavity until the cavity is volumetrically filled.  Thereafter, in the second step, referred to as the packing phase, pressure is maintained on the polymer to permit further flow of polymer into the cavity to compensate for shrinkage as the material solidifies and contracts.  When the component is sufficiently solid, the component may be ejected from the mold.  Both thermoplastic and thermosetting polymers can be injection molded. 
When molding thermoplastics, the temperature of the mold at the cavity surface or wall is maintained at a temperature below the melting temperature of the material to be injected.  As the material flows into the cavity, the liquid material forms a solidified layer along the cavity wall.  This layer may be referred to as the frozen layer and, depending on the processing conditions and the material used, the thickness thereof may vary during filling.  The thickness of the frozen layer is important, because the frozen layer reduces the effective channel width for flow in the cavity and, due to the thermo-rheological characteristics of thermoplastics, typically effects the viscosity of the material flowing thereby. 
Early analytical simulation techniques relied upon two dimensional finite element models, which were found to be beneficial for simulating injection molding of relatively simple, thin-walled components.  More advanced simulation techniques are discussed, for example, in International Patent Application No.  PCT/AU98/00130, assigned to the assignee of the instant invention, the disclosure of which is incorporated herein by reference in its entirety.  In thick or complex components, however, where molten plastic can flow in all directions, traditional thin-wall analytic assumptions that rely on planar regions of specified thickness are not capable typically of predicting this type of flow.  To achieve high accuracy and predictability, a full, three dimensional simulation is desirable to establish, for example, where weld lines will form, air traps will occur, and flow will lead or lag. 
In order to analyze in three dimensions injection molded component designs, it is generally desirable to start with a computerized solid modeling package, such as Pro-Engineer(trademark), CATIA(trademark), I-DEAS(trademark), Solid Works(trademark), Solid Edge(trademark), or other, which is used commonly in mechanical design and drafting applications.  The modeling package can be used to generate three dimensional, photorealistic descriptions of the component geometry, called a solid model.  At present, finite element analysis codes based directly on solid models use nodes to define solid elements such as tetrahedra and hexahedra.  In order to account for the physics involved in injection molding, it is generally desirable to calculate five quantities per node of the finite element model, namely pressure, three orthogonal components of velocity, and temperature.  Given that a suitable model may contain hundreds of thousands of nodes, solution of such complex numerical problems is difficult and requires substantial computer resources. 
U.S. Pat.  No.  5,835,379, issued to Nakano, and related European Patent Application No.  EP 0 698 467 A1, the disclosures of which are incorporated herein by reference in their entirety, suggest a method to reduce the number of variables to be determined in an injection molding finite element model during the filling phase in order to permit calculation using lower computer resources than would be required otherwise.  Nakano discusses the concept of flow conductance, xcexa, to reduce the number of variables to two, namely pressure and flow conductance.  As used herein, the terms flow conductance, fluid conductance, and fluidity are used interchangeably and are to be considered synonymous.  The effect of varying material viscosity is incorporated into the calculation via the flow conductance variable.  This necessarily involves the extrapolation of viscosity data and it is contemplated that this leads to considerable error in the calculation of viscosity.  Also, during the filling and packing phases, the material proximate the cavity walls begins to freeze and the thickness of the layer increases with time until the part is ejected, which is thought to lead to additional error when applying the method of Nakano. 
The viscosity, xcex7, of a polymer melt is frequently measured as a function of temperature and shear rate.  When making a measurement, it is not possible to measure the viscosity at low temperatures approaching the temperature where the material solidifies, because at these low temperatures, the viscosity is relatively high and, at reasonable shear rates, we have discovered that thermal viscous dissipation is significant.  Accordingly, viscosity measurements are taken generally in the range of temperatures at which the melt flows substantially readily.  After measurement, a function is fitted to the measured data.  The available data fitting functions are generally reasonable at high temperatures and hold over a wide range of shear; however, when considering material at the wall that is below or near the solidification temperature, it is necessary to extrapolate the viscosity well beyond the experimental range.  This results in errors in the viscosity value, which in turn creates error in the flow conductance, because viscosity and flow conductance are related by the following relationship:                                           ∇            2                    ⁢          κ                =                  -                      1            η                                              (        1        )            
which can also be represented in the form:
∇xe2x80xa2(xcex7∇xcexa)=xe2x88x921xe2x80x83xe2x80x83(2)
Also, as noted by Nakano, the value of the flow conductance has a small value at the cavity wall and increases with distance away from the wall.  At the wall, the value of flow conductance is close to or equal to zero, because of the very high viscosity there.  A common velocity boundary condition in fluid flow is a zero velocity at the wall.  This equates to a no-slip condition.  When slip occurs, velocity at the wall is non-zero, and can be treated in several ways.  Using the flow conductance approach, the flow conductance can be set to a small, non-zero value.  Nevertheless, as a result, there are, in actuality, variations of flow conductance in the frozen layer which are orders of magnitude less than in the melt.  Such a wide variance in values leads to errors in the numerical scheme of Nakano and others in calculating flow conductance. 
During the packing phase, the values of pressure may also be obtained by solving first for flow conductance.  In packing, we have determined that the frozen layer becomes very significant, to the extent that eventually the majority of nodes are below the solidification temperature.  Hence, the problems alluded to above are exaggerated in the packing phase. 
Conventional three dimensional modeling techniques rely primarily on the underlying laws of conservation of mass and conservation of momentum to predict fluid flow in an injection mold cavity.  However, in order to achieve high accuracy and high predictability in a full, three dimensional simulation of the injection molding process, we have determined conservation of energy principles should be addressed as well.  In order to satisfy all three requirements simultaneously, without consuming inordinate computational resources, creative methods have been developed in accordance with the invention, based upon a variety of new modeling assumptions and methodologies. 
Accordingly, it is one object of the invention to employ conservation of energy principles in modeling fluid flow in a cavity.  The fluid may be a viscous liquid, such as a polymer melt in an injection molding process, or a molten or semi-solid metal used in a casting process.  Because the material also experiences a phase change from the liquid state to a solid state, considering the thermal effects underlying the conservation of energy principles both during initial flow and solidification have been found to be important. 
For example, it has been recognized that during the filling and packing phases, there are three primary heat transfer mechanisms to consider: convection from the incoming melt, conduction out to the mold wall, and viscous dissipation, which is related to the thermal energy produced by shearing within the flowing polymer.  Additionally, there may be other mechanisms, such as compressive heating effects, due to heat generated by compression and cooling resulting from decompression.  All three primary mechanisms contribute significantly to the energy balance during filling.  Accordingly, accurate simulation of injection molding requires typically non-isothermal analysis of the molten polymer as it flows into the mold.  During the packing phase, however, the flow of material is reduced significantly and the main heat transfer mechanisms are conduction to the mold wall and convection of melt from regions of high pressure to regions of lower pressure.  Convection functions quite strongly in the earliest stages of packing, when pressure is first applied.  Note also that packing pressures can be lower or higher than the fill pressure, and can be profiled, that is they can be made to change with time.  Consequently material can flow into the mold and also out of the mold via the feed system, depending on the difference between the pressure in the cavity and the applied pressure at the feed system.  Analysis of the packing phase also typically requires that the melt be considered as a compressible fluid. 
While conventional methods determine flow conductance at a plurality of small elements that divide the region in which fluid flows, according to one embodiment of the present invention, flow conductance is calculated at each of the nodes of each element.  This technique allows a more precise value of the flow conductance to be defined, as the flow conductance may vary in any of a variety of desired fashions from node to node.  The variation will depend on the type of finite element used to discretize the problem.  Use of a nodal value is more desirable than use of an elemental value, as other quantities such as temperature, pressure, and velocity are more precisely defined as point values and, again, may be interpolated from node to node in any fashion.  This interpolation capability also enables the location of the interface between the solid and molten polymer to be accurately determined, which has been found to be particularly relevant in simulation predictive accuracy. 
Accordingly, it is an object of the invention to use a one dimensional analytic function to describe the local temperature distribution at a node.  It is another object of the invention to define the variation of a one dimensional analytic function, with time, to account for heat convection.  It is yet another object of the invention to define the variation of a one dimensional analytic function to account for viscous heat generation.  It is still another object of the invention to define an explicit temperature convection scheme using the one dimensional analytic function.  And it is a further object of the invention to provide anisotropic finite element mesh refinement, including a distance to the cavity wall computation algorithm. 
Moreover, according to one embodiment, the present invention incorporates the concept of a frozen layer to remedy certain of the problems with respect to conventional simulation techniques mentioned above.  A criterion is used to define the interface between the molten polymer and the solidified polymer at the mold cavity wall.  This criterion may be a temperature, a minimum value of velocity, a combination of the two, or some other physical quantity.  In the frozen region, there is no significant movement of polymer and so it is not necessary to calculate the flow conductance there.  Nodes that are determined to be in the frozen region can be removed from the solution process for velocity, pressure, and fluid conductance.  Only temperature need be calculated in the frozen layer.  Note, however, that properties other than temperature can also be computed in the frozen layer.  For example, the material morphology will change with change in state.  Crystallinity of semi-crystalline polymers changes with temperature, and crystals can only be said to form once the melt has cooled sufficiently from the melt state.  The state of stress also depends on temperature and differs between liquid and solid phases.  Polymers are viscoelastic, so their stress state, even in the solid, changes with time and temperature. 
In effect, by only calculating temperature, this reduces the number of variables to be determined to one in the frozen layer and thereby dramatically increases the speed of computation, while reducing the memory required.  By removing the frozen nodes, the need for extrapolation of the viscosity is removed and so the errors associated therewith are removed as well.  Further, by removing nodes in the frozen layer from the analysis, very small values of flow conductance are removed and the equations to be solved are better conditioned than in conventional methods.  Still further, by not calculating fluid conductance in these regions, it has been found that not only is the problem size reduced considerably, but the stability of numerical methods is improved as well. 
Accordingly, it is another object of the invention to determine the position of the solid/liquid interface in the mold cavity and within elements defined in the mesh.  It is still another object of the invention to develop a formulation of elements with a linear variation of material properties throughout.  It is yet another object of the invention to determine an effective viscosity function in elements containing the solid/liquid interface.  It is a further object of the invention to eliminate frozen nodes and elements from the solution domain.  And it is yet a further object of the invention to compute the effective pressure in regions that have solidified. 
One embodiment of the invention involves a method for modeling injection of a fluid into a mold defining a three dimensional cavity, the method including the steps of providing a three dimensional solid computer model defining the cavity, discretizing a solution domain based on the solid model, specifying boundary conditions, solving for filling phase process variables using conservation of mass, momentum, and energy to provide respective filling solutions for at least the portion of the solution domain, then solving for packing phase process variables in a similar manner to provide respective packing phase solutions for at least the portion of the solution domain, and determining whether the filling and packing phase solutions are acceptable.  The filling and packing phase process variables can include density, fluidity, mold cavity fill time, mold cavity packing time, pressure, shear rate, shear stress, temperature, velocity, viscosity, and volumetric shrinkage.  In the event the filling and/or packing phase solutions are unacceptable the boundary conditions and or discretized solution domain can be modified and the analysis repeated until an acceptable result is achieved.  To facilitate a user in determining whether the results are acceptable, a variety of filling and packing phase solutions can be displayed graphically, such as fill time, packing time, density, pressure, shear rate, shear stress, temperature, velocity, viscosity, and volumetric shrinkage. 
Another embodiment of the invention involves a method for modeling injection of a fluid into a mold defining a three dimensional cavity including the steps of providing a three dimensional solid computer model defining the cavity, discretizing a solution domain based on the solid model, specifying boundary conditions, solving for filling phase process variables using conservation of mass, momentum, and energy to provide filling phase solutions for at least some of the portion of the solution domain, and determining whether the solutions are acceptable for injection of the fluid during filling of the mold cavity. 
The discretizing step may include the substep of generating a finite element mesh based on the solid model by subdividing the model into a plurality of connected elements defined by a plurality of nodes.  The mesh generating substep may include generating an anisotropic mesh in thick and thin zones of the model, such that mesh refinement it provides increased resolution in a thickness direction without increasing substantially mesh refinement in a longitudinal direction. 
The boundary conditions may include such parameters as fluid composition, fluid injection location, fluid injection temperature, fluid injection pressure, fluid injection volumetric flow rate, mold temperature, cavity dimensions, cavity configuration, and mold parting plane, and variations thereof. 
Solving for the filling phase process variables may include the substeps of solving for fluidity, pressure, velocity, and viscosity for at least some of the solution domain, where viscosity is based on temperature.  Temperature, in turn, may be based on convective, conductive, and/or viscous dissipation heat transfer contributions.  Velocity and/or viscosity may be calculated iteratively, until pressure converges. 
This method may also include the substep of determining free surface evolution of the fluid in the cavity based on velocity in a given time increment, wherein the free surface evolution is determined iteratively, until the cavity is filled. 
Once the simulation has modeled that the cavity is filled, the method may then solve for packing phase process variables using conservation of mass, momentum, and energy for at least a portion of the solution domain based on respective states of the process variables at termination of filling, to provide respective packing phase solutions for at least some of the solution domain.  A determination can then be made whether the packing phase solutions are acceptable for injection of the fluid during packing of the mold cavity. 
Again, solving for packing phase process variables can include solving for fluidity, pressure, velocity, and viscosity for at least some of the solution domain, where viscosity is based on temperature.  Temperature, in turn, may be based on convective, conductive, and/or viscous dissipation heat transfer contributions.  Velocity and/or viscosity may be calculated iteratively, until pressure converges.  The mass properties of a component produced in accordance with the boundary conditions can also be determined.  The mass properties may include component density, volumetric shrinkage, component mass, and component volume and may also be calculated iteratively, along with velocity and viscosity, until a predetermined pressure profile is completed.