finite elements with increased freedom in choosing shape functions starting with a mathematical statement of the convergence requirements for an element stiffness matrix the paper discusses displacement shape functions that may be used in connection with the potential energy principle in short these functions must be force orthogonal and energy orthogonal but they need not be conforming satisfy interelement compatibility it is shown that the requirements to the displacement functions may be greatly relaxed through slight modifications of the coupling stiffness between fundamental and higher order displacement modes several alternative formulations are examined in particular a new free formulation is suggested using this form which is very simple the only requirement to the displacement patterns used is that they should contain the fundamental deformation modes and be linearly independent applications of the theory to triangular and rectangular plate bending elements are shown the simple stiffness matrix for the latter is given explicitly the numerical results compare favourably with other types of finite elements

optimization of frequencies spectrum in vibrations of flexible structures optimally criteria are derived and used to maximize a set of frequencies for a structure of given weight the criteria include both membrane and bending effects and nonlinear relations between the stiffness and mass matrices an error norm is proposed and used to determine the values of design variables and lagrange multipliers at optimum some details of the accompanying iterative procedure are discussed and the numerical examples of simple test structures are presented

a schur complement formulation for solving free boundary stefan problems of phase change a schur complement formulation that utilizes a linear iterative solver is derived to solve a free boundary stefan problem describing steady state phase change via the isotherm newton approach which employs newton s method to simultaneously and efficiently solve for both interface and field equations this formulation is tested alongside more traditional solution strategies that employ direct or iterative linear solvers on the entire jacobian matrix for a two dimensional sample problem that discretizes the field equations using a galerkin finite element method and employs a deforming grid approach to represent the melt solid interface all methods demonstrate quadratic convergence for sufficiently accurate newton solves but the two approaches utilizing linear iterative solvers show better scaling of computational effort with problem size of these two approaches the schur formulation proves to be more robust converging with significantly smaller krylov subspaces than those required to solve the global system of equations further improvement of performance are made through approximations and preconditioning of the schur complement problem hence the new schur formulation shows promise as an affordable robust and scalable method to solve free boundary stefan problems such models are employed to study a wide array of applications including casting welding glass forming planetary mantle and glacier dynamics thermal energy storage food processing cryosurgery metallurgical solidification and crystal growth

conservative rezoning remapping for general quadrilateral meshes a classic problem in lagrangian numerical hydrodynamics is the conservative transfer of quantities from an old distorted mesh to a new mesh the same problem arises whenever the mesh is changed as for example in adaptive mesh techniques this transfer of information is an interpolation process which is frequently called rezoning or remapping the general problem of conservative rezoning from one arbitrary mesh to another may be formulated as follows m sub k integral integral integral sub v sub k rho r dv that is we compute the mass m sub k of each cell of the new mesh by integrating the known density distribution in the old mesh over the cell volume v sub k a direct intregation is generally prohibitive we show however that is is possible to convert this integral to a surface integral by the appropriate use of the divergence theorem thus greatly reducing the complexity of the problem for two dimensional generall quadrilateral meshes the resulting method is exact and particularly simple

finite element simulation of dynamic wetting flows as an interface formation process a mathematically challenging model of dynamic wetting as a process of interface formation has been for the first time fully incorporated into a numerical code based on the finite element method and applied as a test case to the problem of capillary rise the motivation for this work comes from the fact that as discovered experimentally more than a decade ago the key variable in dynamic wetting flows the dynamic contact angle depends not just on the velocity of the three phase contact line but on the entire flow field geometry hence to describe this effect it becomes necessary to use the mathematical model that has this dependence as its integral part a new physical effect termed the hydrodynamic resist to dynamic wetting is discovered where the influence of the capillary s radius on the dynamic contact angle and hence on the global flow is computed the capabilities of the numerical framework are then demonstrated by comparing the results to experiments on the unsteady capillary rise where excellent agreement is obtained practical recommendations on the spatial resolution required by the numerical scheme for a given set of non dimensional similarity parameters are provided and a comparison to asymptotic results available in limiting cases confirms that the code is converging to the correct solution the appendix gives a user friendly step by step guide specifying the entire implementation and allowing the reader to easily reproduce all presented results including the benchmark calculations

review of options for structural design sensitivity analysis part 1 linear systems the paper reviews options for structural design sensitivity analysis including global finite differences continuum derivatives discrete derivatives and computational or automated differentiation the objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency computational cost and implementation options and effort linear static analysis and transient dynamic analysis are reviewed in a separate appendix special attention is paid to the semi analytical method a future paper will address design sensitivity analysis in nonlinear structural problems

an efficient approximation technique for frequency constraints in frame optimization an efficient approximation technique is presented for dealing with frequency constraints in frame optimization the method is based on a linear approximation of the eigenvalues with respect to the member section properties using this an approximate optimization problem is solved treating the physical dimensions of the beams as design variables when the eigenvalues are required the section properties are first calculated then using the approximation the eigenvalues are estimated and returned to the optimizer when the approximate optimum is found a detailed analysis is performed and the process is repeated examples are presented to demonstrate the method the method is simple yet treats the physical dimensions of the structure as design variables and allows for large move limits during the approximate optimization

optimization of imperfection sensitive symmetric systems for specified maximum load factor a method is presented for optimum design of a symmetric structure which reaches an unstable bifurcation point as the load factor is increased reduction of the maximum load level due to the antisymmetric imperfection is considered and a straightforward algorithm is proposed for calculating the magnitude of reduction of the load factor corresponding to the most critical mode of antisymmetric imperfection the sensitivity coefficients of the bifurcation load factor are calculated by using the interpolation method developed by the authors and an approximate formulation is presented for sensitivity analysis of the maximum load factor it is shown in the examples that the errors in the sensitivity coefficients do not lead to any significant difference in the optimum design

approximate vibration reanalysis of structures thecombinedapproximationsmethodisdevelopedforvibrationreanalysisofstructures inthesolution process the terms of local series approximations are used as basis vectors in global reduced basis approximations the solution steps are straightforward and the method can be readily used with a general e nite element system the computational effort needed by the method is usually much smaller than the effort needed for complete vibration analysis the efe ciency of the calculations and the accuracy of the results can be controlled by the amount of information considered efe cient solutions are obtained by low order approximations and accurate results can be achievedbyconsideringhigher orderterms numericalexamplesshowthataccurateresultsareachievedefe ciently for very large changes in the design nomenclature b matrix dee ned by eq 21 c matrix dee ned by eq 16 e modulus of elasticity i moment of inertia i identity matrix k stiffness matrix kr reduced stiffness matrix l length m mass matrix mr reduced mass matrix r right hand side vector r displacement vector rb matrix of basis vectors s number of basis vectors u upper triangular matrix y vector of coefe cients coefe cient dee ned by eq 35 angle between vectors 1k change in the stiffness matrix 1m change in the mass matrix kronecker delta eigenvalue circular frequency

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

an isochoric domain deformation method for computing steady free surface flows with conserved volumes abstract the domain deformation method has been applied successfully to steady state free surface flows where the volume of the flow domain is unknown v f de almeida gas liquid counterflow through constricted passages ph d thesis university of minnesota minneapolis mn 1995 p a sackinger p r schunk r r rao a newton raphson pseudo solid domain mapping technique for free and moving boundary problems a finite element implementation j comput phys 125 1996 83 103 l c musson two layer slot coating ph d thesis university of minnesota minneapolis mn 2001 however this method does not handle effectively problems where the volume of the flow domain is known a priori this work extends the original domain deformation method to a new isochoric domain deformation method to account for the volume conservation like in the original domain deformation method the unknown shape of the flow domain is mapped onto a reference domain by using the equations of an elastic pseudo solid the difference with the original method is that this pseudo solid is considered incompressible because of the incompressibility the pseudo pressure of the mapping appears as a lagrange multiplier in the equations and it is determined only up to an arbitrary uniform datum by analyzing the coupled fluid flow mapping problem we show that in the finite element setting such pressure datum can be specified by replacing one continuity equation in the fluid domain

truss optimization on shape and sizing with frequency constraints based on orthogonal multi gravitational search algorithm abstract structural optimization on shape and sizing with frequency constraints is well known as a highly nonlinear dynamic optimization problem with several local optimum solutions hence efficient optimization algorithms should be utilized to solve this problem in this study orthogonal multi gravitational search algorithm omgsa as a meta heuristic algorithm is introduced to solve truss optimization on shape and sizing with frequency constraints the omgsa is a hybrid approach based on a combination of multi gravitational search algorithm multi gsa and an orthogonal crossover oc in multi gsa the population is split into several sub populations then each sub population is independently evaluated by an improved gravitational search algorithm igsa furthermore the oc is used in the proposed omgsa in order to find and exploit the global solution in the search space the capability of omgsa is demonstrated through six benchmark examples numerical results show that the proposed omgsa outperform the other optimization techniques

three dimensional shape optimization using fully automatic mesh generation introduction t he technique for associating design variables with mesh data is the most crucial factor in three dimensional shape optimization previously work in three dimensional shape optimization involved specifying design variables by associating parameters directly with grid points on an existing mesh for realistic problems this can be a very tedious and errorprone process in the past shape optimization capabilities have been developed based on a variety of design analysis capabilities ranging from associating parameters with a mesh created manually to associating parameters with control points of a mapped mesh generator special techniques have also been developed to properly move internal grid points during sensitivity calculations more recently a capability based on constructive solid geometry csg has been developed but csg representations are not particularly suitable for design optimization a new optimization approach is demonstrated in this note that uses a new fully automatic mesh generation capability the design model is developed based on design oriented geometric primitives that represent recognizable features of a part and can be assembled into complete solid models that are defined in terms of a small set of design parameters

truss optimization on shape and sizing with frequency constraints an optimality criteria algorithm is presented for three dimentional truss structure optimization with multiple constraints on its natural frequencies both nodal coordinates and element cross sectional areas which are quite different in their natures are treated simultaneously in a unified design space for structural weight minimization first the optimality criterion is developed for a single constraint based on differentiation of the lagrangian function it states that at the optimum all of the variables should have equal efficiencies then the global sensitivity numbers are introduced to solve multiple constraints of frequencies avoiding computation of the lagrange multipliers finally upon the sensitivity analysis the most efficient variables are identified and modified in priority the optimal solution is achieved gradually from the initial design with a minimum weight increment four typical trusses are used to demonstrate the feasibility and validity of the proposed method

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

a new stabilization of adaptive step trapezoid rule based on finite difference interrupts the adaptive step trapezoid rule tr is a generally effective numerical integrator but it is prone to ringing instability and solution stall we introduce a new stabilization algorithm based on finite difference interrupts fdi that reduces ringing and prevents stall unlike previously reported stabilization schemes our algorithm achieves stability and second order accuracy without incurring significant computational cost or spurious diffusion tr with fdi is at least as stable and accurate as existing methods when solving the prototypical scalar problem we demonstrate that it has better stability accuracy and cost when solving the tightly coupled system describing free surface flows of viscoelastic liquids though we demonstrate tr with fdi in the context of a finite element method of lines it is applicable to any tr based algorithm

force method revisited computational mechanics is an important field in engineering with a strong foundation in continuum mechanics and finite element analysis the standard and integrated force methods are reviewed and a structural analysis technique using the finite element force method based on the complementary strain energy is proposed similar to the standard force and the integrated force methods the force method based on the complementary strain energy uses the same equilibrium equations however the compatibility conditions are satisfied through the complementary strain energy the gauss elimination technique has been employed successfully to generate automatically a basis determinate structure and redundant members extension of the force method to dynamics and benchmark problems with displacement stress and frequency constraints are presented

high quality approximation of eigenvalues in structural optimization a new function for approximating natural frequency constraints during structural optimization is presented the rayleigh quotient approximation rqa presented here icreases the quality of the approximate frequency constraint by approximating the modal strain and kinetic energies instead of the frequency eigenvalue itself

composite three dimensional grids generated by elliptic systems a technique is presented for constructing a boundary conforming grid throughout a general threedimensional flow region as a composite of subregion grids each subregion grid is generated numerically by solving a nonlinear system of elliptic equations the boundary values represent nodal points in a quasi twodimensional grid that covers the curved surface bounding the subregion and are generated numerically by a modified elliptic system a primary feature of the technique is that the composite three dimensional grid remains both continuous and smooth across the surface of juncture between any two adjoining subregions sample results are given for a wing body combination

displacement minimization of thermoelastic structures by evolutionary thickness design structural optimization aims at finding an appropriate material distribution to achieve the best structural performance in this paper the evolutionary structural optimization method is further developed to deal with thermoelastic optimization where structural material is progressively redistributed so as to minimize the displacement under thermal and mechanical loading to evaluate the effects of material removal or addition on the displacement of thermoelastic structures a sensitivity number is proposed on the basis of the thickness variation of an element the method proves to be very simple in its mathematical operations and suitable for computer implementation the numerical examples are compared against those by the homogenization method to verify and validate this method of solving thermoelastic optimizations in addition the effect of various temperature distributions on the final optimum structures is investigated in detail

discretization of three dimensional free surface flows and moving boundary problems via elliptic grid methods based on variational principles abstract a new boundary fitted technique to describe free surface and moving boundary problems is presented we have extended the 2d elliptic grid generator developed by dimakopoulos and tsamopoulos 2003 19 and further advanced by chatzidai et al 2009 18 to 3d geometries the set of equations arises from the fulfillment of the variational principles established by brackbill and saltzman 1982 21 and refined by christodoulou and scriven 1992 22 these account for both smoothness and orthogonality of the grid lines of tessellated physical domains the elliptic grid equations are accompanied by new boundary constraints and conditions which are based either on the equidistribution of the nodes on boundary surfaces or on the existing 2d quasi elliptic grid methodologies the capabilities of the proposed algorithm are first demonstrated in tests with analytically described complex surfaces the sequence in which these tests are presented is chosen to help the reader build up experience on the best choice of the elliptic grid parameters subsequently the mesh equations are coupled with the navier stokes equations in order to reveal the full potential of the proposed methodology in free surface flows more specifically the problem of gas assisted injection in ducts of circular and square cross sections is examined where the fluid domain experiences extreme deformations finally the flow mesh solver is used to calculate the equilibrium shapes of static menisci in capillary tubes

structural dynamic frequency response using combined direct and adjoint reduced order approximations a second order combined approximate direct approximate adjoint method for efficient reduced order calculation of frequency response in structural dynamics is presented the method is design oriented it focuses not only on the response itself in the form of displacements and especially stresses but also on sensitivities of the response with respect to structural design variables comparisons of approximate results obtained by mode displacement order reduction reduced order adjoint solutions based on ritz vectors and the second order approximate method demonstrate the accuracy of the second order method and provide lessons that suggest directions for future research

design of frames against buckling using a rayleigh quotient approximation a rayleigh quotient approximation is applied to the design of structures while guarding against elastic instability it approximates buckling eigenvalues by separately estimating the modal strain energy due to the linear and geometric stiffness of the structure previously used during structural optimization for the fundamental natural eigenvalue the rayleigh quotient approximation is derived for the buckling design problem for the first time the critical buckling load is found by solving the eigenvalue problem that arises by considering the geometric nonlinearity of the deforming structure rayleigh s principle is used to justify the choice of intermediate design variables for approximating terms in the rayleigh quotient a truss model illustrates the importance of the design space chosen for approximating the modal strain energy a beamtcolumn two plane frames and a space frame are used to verify the formulation special attention is paid to difficult bimodal optima

structural optimization with frequency constraints this paper presents a design optimization algorithm for structural weight minimization with multiple frequency constraints an optimality criterion method based on uniform lagrangian density for resizing and a scaling procedure to locate the constraint boundary were used in optimization multiple frequency constraints of equality and inequality types were addressed the effectiveness of the algorithm was demonstrated by designing a number of truss structures with as many as 489 design variables no attempt was made to reduce the number of design variables by such procedures as linking and or invoking symmetry conditions the design examples include a 10 bar truss 200 bar truss a modified acoss ii and cofs control of flexible structures mast truss all the structures contain nonstructural mass besides their own mass the algorithm is extremely stable and in all cases the optimum designs were obtained in less than 20 iterations regardless of the size of the structure and the number of design variables

structural optimization with frequency constraints using the finite element force method a structural optimization algorithm is developed to minimize the weight of structures with truss and beam type members under single or multiple frequency constraints the cross sectional areas of the structural members are considered as the design variables the algorithm proposed combines the finite element technique based on the integrated force method with the mathematical programming technique the equilibrium matrix is generated automatically using the finite element analysis and the compatibility matrix is obtained directly using the displacement deformation relations and the single value decomposition technique when combining the equilibrium and the compatibility matrices with the force displacement relations the frequency eigenvalue equations are obtained with element forces as variables three structures composed of truss and frame type members are studied to illustrate the procedure and the results are compared with the literature it is shown that in structural problems with multiple frequency constraints the analysis procedure force or displacement method significantly affects the final optimum design the structural optimization based on the force method results in a lighter design the proposed structural optimization method is efficient to analyze and optimize both truss and beam type structures

optimization of shallow trusses against limit point instability the problem of the maximization of the critical load limit point instability of shallow space trusses of constant volume or weight is considered exact solutions being possible in simple cases a detailed investigation of the question of the uniqueness of the optimum designs for two and four bar shallow truss problems is carried out the optimized trusses satisfy the constant strain energy density criteria and the load maximization problem is verified to be the dual of the minimization of the total potential energy

minimum weight design of truss structures with geometric nonlinear behavior the paper presents an optimization method based on optimality criterion for minimum weight design of structures with geometric nonlinear behavior the nonlinear critical load is determined by finding the load level at which the hessian of the potential energy ceases to be positive definite a recurrence relation based on the criterion that at optimum the nonlinear strain energy density be equal in all the members is used to develop an algorithm sample problems are given to illustrate the application of the method to truss type structures with a large number of design variables number of algorithms based on the optimality criterion approach have been developed to design a minimum weight structure with specified constraints on nodal displacements element stresses system stability etc 1 2 the optimization algorithm consisted of analyzing the structure by the finite element method and using a recurrence relation derived from the appropriate optimality criterion to modify the design variables the optimality criterion was derived by differentiating the lagrangian with respect to the design variables the displacements were assumed to be small and in the finite element analysis linear equilibrium equations were solved to determine the response of the structure to the ap plied loads in the case of system stability the constraints were defined see refs 3 6 by the associated linear eigenvalue problem this definition of system stability for some structures may not be valid because of the nonlinear behavior of the structure which may be due to the geometry of the structure or the presence of geometric imperfections the correct procedure for these structures would be to analyze the structure by using nonlinear equilibrium equations this will be particularly true for large space structures lss in this paper an optimization method based on the optimality criterion approach is presented for structures with geometric nonlinear behavior in the case of a structure optimized with constraints on linear stability the optimum structure can have more than one critical buckling mode this design tends to become im perfection sensitive and a small deviation in the geometry of the structure can reduce its load carrying capacity sub stantially the imperfection sensitivity of the optimized structure can be reduced by designing the structure so that buckling loads associated with the critical buckling modes are not equal see ref 7 a real structure has geometric im perfections and these structures tend to have nonlinear

assessing the quality of curvilinear coordinate meshes by decomposing the jacobian matrix an algebraic decomposition of the jacobian matrix j x i email protected j which relates physical and computational variables is presented this invertible decomposition parameterizes the mesh by the physically intuitive qualities of cell orientation cell orthogonality cell volume and cell aspect ratio this decompostion can be used to analyze numerically generated curvilinear coordinate meshes and to assess the contribution of the mesh to the truncation error for any specific differential operator and algorithm this is worked out in detail for laplace s equation in nonconservative and conservative forms an analysis by g h k 1 of the mesh contribution to truncation error for the full potential code tair is given in abbreviated form the variables introduced here and their derivatives are also natural lagrange multipliers for adaptive mesh algorithms based on a variational principle

algebraic grid generation three methods are described for transforming grids in bounded two and three dimensional physical domains into a uniform grid in a rectangular computational domain the methods are based on mathematical interpolation functions and do not require the solution of differential equations or the use of complex variables they are simply referred to as algebraic methods and are called transfinite interpolation the multisurface method and the two boundary technique the primary advantage of the methods is that they provide explicit control of physical grid shape and physical grid spacing secondly they require relatively few computations consequently the application of inter active computer graphics in conjunction with the methods is advocated for rapid generation of grids the basic mathematical structure of each method is described and particular attention is given to surface representation parameter variable generation and control function generation physical boundary topology and grid derivative requirements are presented

a quasi elliptic transformation for moving boundary problems with large anisotropic deformations we have developed a quasi elliptic set of equations for generating a discretization mesh that optimally conforms to an entire domain that undergoes large deformations in primarily one direction we have applied this method to the axisymmetric problem of the transient displacement of a viscous liquid by a high pressure gas the liquid initially fills completely a tube the diameter of which may be constant or change smoothly suddenly or periodically along its finite length key ingredients for the success of the proposed transformation are limiting the orthogonality requirements on the mesh and employing an improved node distribution function along the deforming boundary the retained orthogonal term along with the penalty method for the imposition of the boundary conditions overcome the inherent restrictions of a conformal transformation producing meshes of high quality this term also eliminates the discontinuous slopes of the coordinate lines that are normal to the free surface these usually arise due to the harmonic transformation around highly deforming surfaces the mathematical anisotropy in the mesh generating equations directly corresponds to the physical one when the initially straight and normal to the axis of symmetry air fluid interface at the tube entrance develops mostly along the axial direction generating a long open bubble moreover the generalized node distribution imposes an optimal discretization of the bubble surface itself combining this scheme with the mixed finite element method produces a powerful tool with extended robustness and accuracy in order to reduce the computational cost the resulting set of equations is solved with a non linear gauss seidel technique and a variable time step specific applications of the proposed scheme are presented

application of a variational method for generating adaptive grids the application of variational methods for generating adaptive grids is not as straightforward as one is led to believe proper scaling suitable weight functions and appropriate clustering on boundaries must be employed to obtain a satisfactory grid this work which is based on the framework developed by brackbill and saltzman provides simple methods for determining scaling and investigates possible options for selecting the weight function and clustering points on the boundaries the concepts developed here are applied to two two dimensional problems a model problem based on burger s equation which contains two length scales and transonic flow past airfoils using the euler equations

tracking birth of vortex in flows in order to avoid undesired effects from vortices in many industrial processes it is important to know the set of operating parameters at which the flow does not have recirculation the map of these conditions in the parameter space is called vortex free operating window here we propose an efficient way to construct such window automatically without expensively checking every possible flow states the proposed technique is based on tracking a path in the parameter space at which the local kinematic condition at a stagnation point for vortex birth is satisfied this multiparameter continuation is performed by solving an augmented navier stokes system in the augmented system the birth condition and the governing equations was represented in galerkin s finite element context we used the proposed method in two important coating flows with free surfaces single layer slot coating and forward roll coating

combined direct adjoint reduced order approximations for design oriented structural acoustics the accuracy of modal order reduction methods in structural dynamics and structural acoustics depends on the particular problem analyzed and particular modal bases selected modal order reduction methods can become quite inaccurate when sensitivities of response functions with respect to design variables are required past research in linear aerodynamics and structural dynamics had shown how direct and adjoint solutions can be combined to yield reduced order analysis and sensitivity approximations of the problems at hand with second order accuracy efforts are reported to extend the methodology to structural acoustic problems of fluid filled cavities enclosed by flexible walls in this case stress and pressure response to excitation has to be evaluated over a range of frequencies improved and more consistent accuracy is reported over a range of frequencies with the second order method when accuracy of both approximate direct and approximate adjoint solutions is comparable the second order method cannot improve accuracy of first order methods in general when one of the approximate first order solutions is highly inaccurate

reformulation of gray s duality for attractive spin systems and its applications duality relations which associate two markov processes in different state spaces have been useful tools in the study of the long term behaviour of the stochastic processes of spin systems in 1986 gray introduced a new duality theory which is applicable to general spin systems with attractive transition rates the theory was developed by making full use of graphical representations in the present paper gray s duality is reformulated by studying the action of the formal generators on a newly chosen duality function and the dual processes defined in the state space upsilon a a is a finite subset of y are discussed where y is a collection of finite subsets of zd as applications of the submodularity of the survival probability sigma a of the dual processes rigorous lower bounds of the critical values are derived for the theta contact process the multi particle creation model and the sexual reproduction process

stochastic spatial models in the models we will consider space is represented by a grid of sites that can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites our main aim here is to explain an idea of durrett and levin 1994 the behavior of these models can be predicted from the properties of the mean field ode i e the equations for the densities of the various types that result from pretending that all sites are always independent we will illustrate this picture through a discussion of eight families of examples from statistical mechanics genetics population biology epidemiology and ecology some of our findings are only conjectures based on simulation but in a number of cases we are able to prove results for systems with fast stirring by exploiting connections between the spatial model and an associated reaction diffusion equation

flows in forward deformable roll coating gaps roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter a continuous liquid layer and to apply it to a continuous flexible substrate of the two rolls that make a forward roll coating gap one is often covered by a layer of deformable elastomer thin films can be obtained without the risk of clashing two hard rolls liquid carried into the converging side of the gap can develop high enough pressure to deform the resilient cover which changes the gap geometry and thus alters the velocity and pressure fields the complete understanding of the flow in this situation is vital to the optimization of this widely used coating method however this elastohydrodynamic action is not well understood the situation is similar to what is called the soft elastohydrodynamic lubrication regime soft ehl however the range of minimum distance between the rotating rolls roll speed and therefore flow rate through the gap in the roll coating process is one to three orders of magnitude larger than the typical values reported in previous work on soft ehl earlier works on deformable roll coating analyzed the action with both the lubrication approximation and the full navier stokes solution and different one dimensional models of roll cover deformation in order to test the accuracy of the past approaches and to evaluate the relationship between the empirical constant used in the one dimensional model to the relevant physical parameters a complete two dimensional formulation has to be employed for both the liquid flow and the solid deformation in this work the flow between a rigid and a deformable rotating roll was examined by solving the complete navier stokes system coupled with a non linear plane strain model of the roll cover deformation the approximate and computationally cheaper approach is evaluated in which the compliant wall is represented by an array of radially oriented hookean springs the equation system was solved by the galerkin finite element method the resulting set of non linear algebraic equations of the fully coupled problem was solved by newton s method with initialization by pseudo arc length continuation as parameters were varied results show how roll deformation affects the total flow rate and forces on the rolls and illustrate how a deformable roll can be used to obtain thin coated layers with much less sensitivity to roll runout than those obtained with rigid rolls

a direct application of higher order parametric programming techniques to structural optimization computational methods based on a sequence of parametric programming problems are presented for solving constrained optimization problems cop without any parameter an auxiliary parametric programming problem appp is formulated in order to solve cop the procedure is started with an arbitrary initial solution which is the trivial solution of appp corresponding to the initial value of the parameter then the optimal solution for the final value of the parameter which is the optimal solution of cop is estimated by taylor s expansion with respect to the parameter where higher order terms are incorporated it is shown that the incorporation of the higher order terms indeed leads to a faster convergence of the solution as an extension of the method a general algorithm is presented for optimum design problems with state variable constraints which are implicit functions of the design variables logarithmic penalty functions are incorporated and the weight coefficients for the penalty terms are updated continuously the derivatives of the state variables with respect to the parameter and their sensitivity coefficients are expressed explicitly in terms of those of the design variables finally a method of simultaneous analysis and optimization is developed for trusses with geometrical non linearity

optimality criterion techniques applied to frames having general cross sectional relationships an optimality criterion based method developed in this research is capable of locating the optimal design of wf steel frames with members having general cross sectional relationships among the area moment of inertia and section modulus the method uses the idea of one most critical constraint to avoid the calculation of large sets of lagrange multipliers and also avoids the scaling procedure normally used in other optimality criterion techniques several example structures designed under stress displacement and minimum area constraints demonstrate the efficiency reliability and simplicity of the method although the method of this research uses the idea of one most critical constraint the optimal designs presented in this paper have several active constraints

a physics based parameterization method for shape optimization a physical approach is presented to provide shape modification of nurbs curve the movement of the control polygon of nurbs is simulated by a physical frame structure and the curve deformation is then carried out through changes of the mode coordinates which are derived from the eigenvectors of the finite element equations of the physical frame system the shape of the design boundaries is described by a linear combination of the mode shape vectors based on the presented design variable representation and p version finite element analysis an integrated approach to structural shape optimization is discussed a method is developed that enables the description of the shape to be adaptively refined during the shape optimization process numerical examples are used to demonstrate the applicability of this approach to structural shape optimization

computational method for optimization of structural shapes in the direct method of shape optimum design mesh regridding is a necessary step in avoiding the distortion of finite element grids the effect of internal nodal movements on the shape design sensitivity associated with the regridding process has been carefully examined in this paper it shows that to maintain computational accuracy the effects of internal nodal movement should be considered in shape design sensitivity analysis to take the internal nodal movement into account the numerical accuracy of the speed method is investigated a new approach which constructs the velocity field over the element domain is presented although it takes more computational time the new approach does improve the accuracy of shape design sensitivity

alternative formulations for structural optimization an evaluation by using trusses the concept of simultaneous analysis and design sand is revisited with two objectives in mind 1 to study and evaluate three alternative formulations for structural optimization using trusses as examples and 2 to implement the formulations with a structural analysis program and study their implementation aspects the idea is to gain insights for use of the alternative formulations for structural optimization three alternative formulations are defined using displacements and forces or stresses as optimization variables all of the problem functions in the formulations become explicit in terms of the variables gradients of the functions can be evaluated more easily compared to the conventional approach where special design sensitivity analysis methods are needed existing analysis software is integrated with an optimizer based on a sparse sequential quadratic programming method to solve examples with known solutions with the sand type formulations only the pre and postprocessing capabilities of the analysis software are used to evaluate the problem functions however the conventional formulation also needs the equation solving capabilities for analysis as well as design sensitivity analysis performance of the formulations is evaluated with extensive numerical experiments and their advantages and disadvantages are discussed based on the present research it is concluded that the alternative sand formulations are more efficient compared to the conventional approach and have potential for further development for practical applications particularly the formulations that do not require calculation of the global stiffness matrix

optimum design of structures with both natural frequency and frequency response constraints an efficient method is developed to compute the sensitivities of eigenvalues eigenvectors and frequency responses in a reduced approximate rnodel the reanalyses of a modified structure for those dynamic responses are also derived in the approximate model an optimization process utilizing the efficiently calculated sensitivities and reanalyses of the modified structure is formed the difficulties of finding eigenvector sensitivities for repeated modes have also been addressed and averted in the approximate model two numerical examples show satisfactory outcomes by using this proposed approach

discretization of free surface flows and other moving boundary problems a system of elliptic partial differential equations and boundary conditions has been developed for generating boundary fitted element discretizations of two dimensional free and moving boundary problems terms in the differential equations are scaled for dimensional homogeneity and adjustable weighting of orthogonality smoothness and concentration of the coordinate mesh they govern grid points become finite element nodes mapped isoparametrically or subparametrically from a simple or patched computational domain concentration terms contain control functions and parameters that influence node spacing along each coordinate independently overall control is by patchwise parameters and functions successful selection of these to follow deforming flow regions is straightforward and is illustrated by analysis of steady and transient slide coating flows

a new scheme for efficient and direct shape optimization of complex structures represented by polygonal meshes in this paper a new shape optimization approach is proposed to provide an efficient optimization solution of complex structures represented by polygonal meshes our approach consists of three main steps 1 surface partitioning of polygonal meshes 2 generation of shape design variables on the basis of partitioned surface patches and 3 gradient based shape optimization of the structures by reducing a weighted compliance among all load cases the main contributions of this paper include i that our approach can be directly applied on polygonal meshes with the reduction of design variables or decision variables by 10 to 1000 times compared to the conventional design variable scheme of using each mesh node ii our perturbation scheme is mathematically proven with respect to maintaining the smoothness of each surface patch except on its boundary and iii overall our approach can be used to automate time consuming shape optimization of polygonal meshes to a greater extent numerical experiments have been conducted and the results indicate the effectiveness of the approach copyright 2003 john wiley sons ltd

optimization of geometrically nonlinear thin shells subject to displacement and stability constraints a methodology is developed for shape optimization of thin plate and shell structures undergoing large deflections subject to displacement and system stability constraints the optimization method considers shape parameters and overall thickness of the structure as the design variables and aims to minimize the total mass of the structure subject to stability or displacement constraint two optimality criteria based on karush kuhn tucker conditions are developed for mass minimization problems optimality criteria are combined with nonlinear corotational analysis to optimize structures with geometric nonlinearity the method is applied to plate and shallow shell structures the efficiency of the developed design optimization methodology is compared with that of the gradient based method of optimization sequential quadratic programming

nonlinear analysis of optimized structure with constraints on systemstability an optimization algorithm based on an optimality criterion was used to design a minimum weight space truss with different constraint requirements on system stability the constraints were specified so that the eigenvalues associated with all the critical buckling modes are either equal or separated by a specified factor for the second case the critical buckling mode was preselected from all the possible critical modes the designs obtained for the various constraint conditions were analyzed with and without specified geometric imperfections using a nonlinear finite element program which accounts for geometric nonlinearity the results obtained for various designs were compared for their imperfection sensitivity

marangoni benard convection with a deformable free surface computations of marangoni convection are usually performed in two or three dimensional domains with rigid boundaries in two dimensions allowing the free surface to deform can result in a solution set with a qualitatively different bifurcation structure we describe a finite element technique for calculating bifurcations that arise due to thermal gradients in a two dimensional domain with a deformable free surface the fluid is assumed to be newtonian to conform to the boussinesq approximation and to have a surface tension that varies linearly with temperature an orthogonal mapping from the physical domain to a reference domain is employed which is determined as the solution to a pair of elliptic partial differential equations the mapping equations and the equilibrium equations for the velocity pressure and temperature fields and their appropriate nonlinear boundary conditions are discretized using the finite element method and solved simultaneously by newton iteration contact angles other than 90 degrees are shown to disconnect the transcritical bifurcations to flows with an even number of cells in the expected manner the loss of stability to single cell flows is associated with the breaking of a reflectional symmetry about the middle of the domain and therefore occurs at a pitchfork bifurcation point for contact angles both equal to and less than 90 degrees

effective two point function approximation for design optimization anew two pointapproximationofafunctionthatisverystraightforwardtobuildandthatprovidesaccurateand stable approximation is presented earlier developments of the two point approximations had either incomplete matches at two data points or needed the solution of additional equations to get all of the parameters the present two point approximation is an incomplete second order taylor series expansion in terms of intervening variables thehessianmatrixhasonlydiagonalelements anditdependsondesignvariables theexponentofeachintervening design variable and the unknown constant of the second order terms are evaluated by matching the derivatives and the value of the approximation with the previous data point gradients and the value of the original function respectively all of theunknownsareidentie ed in aclosedform both thefunctionandthegradientof thetwo point approximation are equal to those of the original function at two data points several examples are given to show the accuracy and efe ciency of this method

a newton raphson pseudo solid domain mapping technique for free and moving boundary problems an implicit pseudo solid domain mapping technique is described that facilitates finite element analysis of free and moving boundary problems the technique is based on an implicit full newton strategy free of restrictions on mesh structure this leads to many advantages over existing domain mapping techniques the fully coupled approach using newton s method is particularly effective for problems with strong coupling between the internal bulk physics and the governing physics at unknown free boundary locations it is also useful when the distinguishing conditions which constrain the free boundary shape provide only an implicit dependence on the boundary location unstructured meshes allow for efficient resolution of internal and boundary layers and other regions of strong local variations in the solution and they also reduce the amount of user interaction required to define a problem since the meshes may be generated automatically the technique is readily applied to steady or transient problems in complex geometries of two and three dimensions examples are shown that include free and moving boundary problems from solidification and capillary hydrodynamics

multimodal optimization of structures with frequency constraints optimally designed structures under eigenvalue constraints natural frequency or buckling loads are likely to display multiple eigenvalues this introduces singularity of eigenvalue derivatives with respect to the design vector which does not allow use of the kuhn tucker conditions the paper presents the optimality criteria which takes that singularity into account and in contrast to commonly used kuhn tucker conditions they are valid for multiple eigenvalues the form of that criteria is more general than the one derived from the kuhn tucker conditions an algorithm is presented which applies those criteria and also uses a new method of computation of lagrange multipliers the algorithm allows for an arbitrary number of eigenvectors to be considered in optimality criteria and the actual modality of the problem is determined automatically examples of optimization of truss structures modelled by finite element method are included

optimal structural design with damping constraint limitations the algorithm optimization presented uses the optimality criteria method so as to minimize the total weight of the structure in the presence of a constraint of the damping type the goal of this study is to increase the structural damping by optimizing the viscoelastic and elastic material distribution over the structure the types of damping considered are hysteristic the gradient of the constraint function in relation to the design variables is calculated with the aid of a complex variable sensitivity method several numerical results are presented to validate the algorithm proposed

optimum design of structures with multiple constraints automated design of large aerospace structures requires efficient optimization algorithms because of a large number of design variables and design constraints most of the difficulties associated with large structural design are solution convergence and computer resources requirements practical aerospace structures generally involve limitations on gage sizes displacements stresses and frequencies the objective of this study was to determine what techniques are reliable and efficient for optimization of a complex design problem the study examined the relative numerical performance of various optimization methods as candidates for a hybrid algorithm using optimality criteria and mathematical programming methods several optimization programs were used to design trussand wing type structures and the iteration history for each technique is presented the computer programs using the method of feasible directions and a modified newton s method of unconstrained minimization were the most reliable mathematical programming methods

coordinate system control adaptive meshes the discussion of the formulation and properties of the adaptive mesh algorithm is as complete as space permits as should be clear from the discussion adaptive gridding is most useful for singular perturbation problems where localized regions with large gradients develop spontaneously often choosing the correct weight function is not trivial and sometimes it is appropriate to cause the mesh to respond to structure in several dependent variables simultaneously

aeroservoelastic aspects of wing control surface planform shape optimization equivalent plate structural modeling and doublet point lifting surface unsteady aerodynamics are used to obtain analytic sensitivities of aeroelastic and aeroservoelastic response with respect to wing and control surface planform shape parameters rational function approximations for unsteady aerodynamic forces their shape sensitivities and the resulting linear time invariant state space models of aeroservoelastic systems and their shape sensitivities are examined the goal is to develop effective and numerically efficient approximation techniques for wing shape optimization for use with nonlinear programming and approximation concepts as a multidisciplinary optimization strategy effects of structural and unsteady aerodynamic modeling errors are studied examination of approximation accuracy using alternative approximation techniques and the resulting move limits provide insight and experience on the way to realistic wing control surface shape optimization with active controls and aeroservoelastic constraints

generation of orthogonal and nearly orthogonal coordinates with gridcontrol near boundaries a numerical procedure is presented for the generation of boundary fitted curvilinear coordinates with controllable mesh spacing and orthogonality the technique is based on a straightforward two step procedure involving the solution of poisson s equation the method is capable of generating orthogonal grids with partial control of the mesh spacing or nearly orthogonal grids with strict control of the mesh spacing the inhomogeneous terms in the poisson equations are automatically adjusted during the grid generation the technique is applicable to two dimensional simply connected regions typical of airfoils cascades diffusers and inlets

on the elliptic mesh generation in domains containing multiple inclusions and undergoing large deformations we present an improved method to generate a sequence of structured meshes even when the physical domain contains deforming inclusions this method belongs to the class of arbitrary lagrangian eulerian ale methods for solving moving boundary problems its tools are either a separate mappings of the domain boundaries and enforcing the node distribution on lines emanating from singular points or b domain decomposition and separate mappings of each subdomain using suitable coordinate systems the latter is shown to be more versatile and general in both cases a set of elliptic equations is used to generate the grid extending in this way the method advanced by dimakopoulos and tsamopoulos y dimakopoulos j a tsamopoulos a quasi elliptic transformation for moving boundary problems with large anisotropic deformations j comput phys 192 2003 494 522 we shall present examples where this earlier method and all other mesh generating methods which are based on a conformal mapping or solving a quasi elliptic set of pdes fail to produce an acceptable mesh and accurate solutions in such geometries furthermore in contrast to other methods appropriate boundary conditions and constraints such as orthogonality of specific mesh lines and prespecified node distributions on them can be easily implemented along a specific part of the domain or its boundary hence no attractive terms at specific corners or singular points are needed to increase the mesh resolution around the moving interfaces while keeping low the memory requirements and the computational time a local mesh refinement technique has been incorporated as well the method is demonstrated in two challenging examples where no remeshing is required in spite of the large domain deformations in the first one the transient growth of two bubbles embedded in a viscoelastic filament undergoing stretching in the axial direction is examined while in the second one the linear and non linear dynamics of two bubbles in a viscous medium are determined in an acoustic field the large elasticity of the filament in the first case or the large inertia in the second case coupled with the externally induced large deformations of the liquid domain requires the accurate calculation which is achieved by the method we propose herein the governing equations are solved using the finite element galerkin method with appropriate modifications to solve the hyperbolic constitutive equation of a viscoelastic fluid these are coupled with an implicit euler method for time integration or with arnoldi s algorithm for normal mode analysis

on the discrete variant of the traction method in parameter free shape optimization abstract one of the major challenges in the parameter free approach to computational shape optimization is the avoidance of oscillating i e non smooth boundaries in the optimal design trials in order to achieve this we investigate a method for regularization that corresponds to the discrete variant of the so called traction method in this approach the design updates are generated in terms of a displacement field which is obtained as the solution to an auxiliary boundary value problem that is defined on the actual design domain the main idea herein is to apply fictitious nodal forces corresponding to the discrete sensitivity of the objective function we propose an algorithm in which constraint functions will be taken into account by using an augmented lagrangian formulation and a step length control ensures a sufficient decrease condition in terms of the objective function within each iteration we examine the benefits of the proposed regularization method on the basis of some numerical examples in comparison to an unregularized steepest descent algorithm

structural design optimization of shallow frame structures undergoing large deflections an optimization algorithm for structural design against instability is developed for shallow beam structures undergoing large deflections the algorithm is based on the maximization of the limit load under specified volume constraint the analysis for obtaining the limit load involves coupling of axial and bending deformations and is based on the nonlinear finite element analysis using the displacement control technique the optimization is carried out using both the sequential quadratic programming sqp and optimality criterion oc techniques and the results are compared for the sqp technique the sensitivity derivatives of the critical load factor are calculated using the adjoint method based on the information obtained from the nonlinear buckling analysis a shallow plane arch illustrates the structural design optimization methodology and the results are compared with those in the literature it is shown that a design based on the generalized eigenvalue problem linear buckling gives an optimum limit load less than the initial limit load whereas the optimization using the nonlinear buckling analysis obtains a larger value for the optimum limit load compared to the initial limit load it has also been demonstrated that the optimum results obtained using oc technique are in good agreement with those obtained through sqp technique however the computational time for the oc is significantly lower than that of sqp which requires search techniques and sensitivity of the limit load for its successful completion and termination

a study of optimal membrane triangles with drilling freedoms abstract this article compares derivation methods for constructing optimal membrane triangles with corner drilling freedoms the term optimal is used in the sense of exact inplane pure bending response of rectangular mesh units of arbitrary aspect ratio following a comparative summary of element formulation approaches the construction of an optimal three node triangle using the andes formulation is presented the construction is based upon techniques developed by 1991 in student term projects but taking advantage of the more general framework of templates developed since the optimal element that fits the andes template is shown to be unique if energy orthogonality constraints are enforced two other formulations are examined and compared with the optimal model retrofitting the conventional linear strain triangle element by midpoint migrating and congruential transformations is shown unable to produce an optimal element while rank deficiency is inevitable use of the quadratic strain field of the 1988 allman triangle or linear filtered versions thereof is also unable to reproduce the optimal element moreover these elements exhibit serious aspect ratio lock these predictions are verified on benchmark examples

on one dimensional stretching functions for finite difference calculations abstract the class of one dimensional stretching functions used in finite difference calculations is studied for solutions containing a highly localized region of rapid variation simple criteria for a stretching function are derived using a truncation error analysis these criteria are used to investigate two types of stretching functions one is an interior stretching function for which the location and slope of an interior clustering region are specified the simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine it was first employed by thomas et al aiaa j 10 1972 887 the other type of function is a two sided stretching function for which the arbitrary slopes at the two ends of the one dimensional interval are specified the simplest such general function is found to be one based on the inverse tangent the special case where the slopes were both equal and greater than one was first employed by roberts the general two sided function has many applications in the construction of finite difference grids examples of such applications are found in the listed references

shape optimal design using fictitious loads implementation aspects of the natural velocity field approach for shape optimal design are discussed the problem of concern is to find the optimum shape of an elastic body that requires minimizing an objective function subject to stress displacement frequency and manufacturing constraints in this paper fictitious loads are applied at certain control nodes on an auxiliary structure and the deformation produced by these loads is used to update the shape the use of beam stiffeners in the auxiliary structure leads to smoother shapes furthermore softer material properties in the auxiliary structure result in faster convergence when iterating through the feasibile region the method is applied in conjunction with both discrete and continuum methods of design sensitivity analysis

generalized compound scaling algorithm and application to minimum weight design of plate structures this paper presents the generalized compound scaling algorithm the optimum designs are reached by simply scaling the design variables to an optimum intersection of multiple constraints a four noded isoparametric plate element is used for modeling the structure

optimum design of nonproportionally damped structures using method of optimality criterion a structural optimization problem that deals with nonproportional damping is discussed in this paper the damped motion equation is transformed into a state space form with displacements and velocities as state variables the transformed system is in the same form as an undamped eigenvalue problem except that the system matrices are not positive definite taking advantage of this familiar eigensystem the damped eigenvalues and eigenvectors can be obtained using widely available computer codes the sensitivities for the damped eigenvalues can also be analytically calculated the structural optimization of the nonproportionally damped system is completed by using the method of optimality criterion two numerical examples demonstrate this proposed approach

inverse eigenvalue problem in structural design a method is presented for the design of a structure with specified low order natural frequencies the method can also be used for design problems in which certain bands devoid of any frequency constraints are specified this can further be used to generate initial feasible designs for optimum design problems with frequency constraints

structural design optimization of nonlinear symmetric structures using the group theoretic approach among multidisciplinary analysis and optimization problems structural optimization of geometrically nonlinear stability problems is of great importance especially in structures used in space applications because of their long and slender configurations in this study application of the group theoretic approach gta in structural optimization of geometrical nonlinear problems under system stability constraint has been investigated according to gta the number of displacement degrees of freedom in the initial configuration can be reduced significantly by using the set of symmetry transformations of the undeformed structure to construct a projection matrix from full space to a reduced subspace spanned by the symmetry modes a structural optimization algorithm is developed for shallow structures undergoing large deflections subject to system stability constraint the method combines the nonlinear buckling analysis based on the displacement control technique using gta with the optimality criteria approach based on the potential energy of the system a shallow dome truss structure has been designed to illustrate the proposed methodology this paper demonstrates that structural optimization of nonlinear symmetric structures using gta is computationally efficient and excellent agreement exists between optimal results in full space and those in the reduced subspace also it is shown that structural design based on the generalized eigenvalue problem linear buckling highly underestimates the optimum mass which may lead to structural failure