Problems In Computational Mechanics Research Articles

Convergence of adaptive finite element methods in computational mechanics

The a priori convergence of finite element methods is based on the density property of the sequence of finite element spaces which essentially assumes a quasi-uniform mesh-refining. The advantage is guaranteed convergence for a large class of data and solutions; the disadvantage is a global mesh refinement everywhere accompanied by large computational costs. Adaptive finite element methods (AFEMs) automatically refine exclusively wherever their refinement indication suggests to do so and consequently leave out refinements at other locations. In other words, the density property is violated on purpose and the a priori convergence is not guaranteed automatically and, in fact, crucially depends on algorithmic details. The advantage of AFEMs is a more effective mesh in many practical examples accompanied by smaller computational costs; the disadvantage is that the desirable convergence property is not guaranteed a priori. Efficient error estimators can justify a numerical approximation a posteriori and so achieve reliability. But it is not theoretically justified from the start that the adaptive mesh-refinement will generate an accurate solution at all. In order to foster the development of a convergence theory and improved design of AFEMs in computational engineering and sciences, this paper describes a particular version of an AFEM and analyses convergence results for three model problems in computational mechanics: linear elastic material (A), nonlinear monotone elastic material (B), and Hencky elastoplastic material (C). It establishes conditions sufficient for error-reduction in (A), for energy-reduction in (B), and eventually for strong convergence of the stress field in (C) in the presence of small hardening.

Monge–Ampere grids and the multidimensional mapped Fourier method

The efficiency of a numerical method can be greatly improved by combining it with coordinate transformations tailored to a given problem. This is the basis for the mapped Fourier methods. However, obtaining "good" coordinate transformations is a major obstacle for this approach in multidimensions. Here, we calculate coordinate transformations based on solving the Monge-Ampere equation. These transformations are combined in the mapped Fourier method and applied to Schrodinger's equation in multidimensions. Dramatic improvements in accuracy compared to the standard Fourier method were observed in eigenvalue calculations for two-dimensional systems. This work indicates that the Monge-Ampere equation may serve as a useful tool for constructing efficient representations for problems in computational quantum mechanics and other fields.

Singular and Regular Implementations of the Hybrid Boundary Node Method

The hybrid boundary node method (HdBNM) combines a modified function with the moving least squares approximation to form a boundary-only truly meshless method. This paper describes two implementations of the HdBNM, the singular hybrid boundary node method (ShBNM) and the regular hybrid boundary node method (RhBNM). The ShBNM and RhBNM were compared with each other, and the parameters that influence their performance were studied in detail. The convergence rates and their applicability to thin structures were also investigated. The ShBNM and RhBNM are found to be very easy to implement and to efficiently obtain numerical solutions to computational mechanics problems.

Three-dimensional radiative transfer modeling using the control volume finite element method

In the present study, a three-dimensional algorithm for the treatment of radiative heat transfer in emitting, absorbing and scattering media is developed. The approach is based on the utilization of control volume finite element method (CVFEM) which, to the knowledge of the authors, is applied at the first time to 3D radiative heat transfer in participating media. The accuracy of the present algorithm is tested by comparing its predictions to other published works. Comparisons show that CVFEM produces good results. Moreover, this approach permits compatibility with other numerical methods used for computational fluids mechanics problems.

Semidefinite optimization models for limit and shakedown analysis problems involving matrix spreads

Limit and shakedown analysis problems of Computational Mechanics lead to convex optimization problems, characterized by linear objective functions, linear equality constraints and constraints expressing the restrictions imposed by the material strength. It is shown that two important strength criteria, the Mohr–Coulomb and the Tresca criterion, can be represented as systems of semidefinite constraints, leading this way to semidefinite programming problems.

Extensions of goal-oriented error estimation methods to simulations of highly-nonlinear response of shock-loaded elastomer-reinforced structures

This paper describes extensions of goal-oriented methods for a posteriori error estimation and control of numerical approximation to a class of highly-nonlinear problems in computational solid mechanics. An updated Lagrangian formulation of the dynamical, large-deformation response of structures composed of strain-rate-sensitive elastomers and elastoplastic materials is developed. To apply the theory of goal-oriented error estimation, a backward-in-time dual formulation of these problems is derived, and residual error estimators for meaningful quantities of interest are established. The target problem class is that of axisymmetric deformations of layered elastomer-reinforced shells-of-revolution subjected to shock loading. Extensive numerical results on solutions of representative problems are given. It is shown that extensions of the theory of goal-oriented error estimation can be developed and applied effectively to a class of highly-nonlinear, multi-physics problems in solid and structural mechanics.

Nitsche's method for interface problems in computa‐tional mechanics

AbstractWe give a review of Nitsche's method applied to interface problems, involving real or artificial interfaces. Applications to unfitted meshes, Chimera meshes, cut meshes, fictitious domain methods, and model coupling are discussed. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

PROBLEMS OF COMPUTATIONAL MECHANICS RELATE TO FINITE-ELEMENT ANALYSIS OF STRUCTURAL CONSTRUCTIONS

The problems of computational mechanics, concerning with application of finite element analysis to structural constructions, are discussed. Our attention is addressed to medium-class software for personal computers with which structural constructions are usually analyzed. The complexity of a system and simplicity of its components, the large-scale of finite element problem, the heterogeneity of finite elements and its coupling, the estimation of correctness of finite element model, the problems of seismic analysis, the problem of indeterminacy and so on are the objective of this work.

DSC‐Ritz method for high‐mode frequency analysis of thick shallow shells

AbstractThis paper addresses a challenging problem in computational mechanics—the analysis of thick shallow shells vibrating at high modes. Existing methods encounter significant difficulties for such a problem due to numerical instability. A new numerical approach, DSC‐Ritz method, is developed by taking the advantages of both the discrete singular convolution (DSC) wavelet kernels of the Dirichlet type and the Ritz method for the numerical solution of thick shells with all possible combinations of commonly occurred boundary conditions. As wavelets are localized in both frequency and co‐ordinate domains, they give rise to numerical schemes with optimal accurate, stability and flexibility. Numerical examples are considered for Mindlin plates and shells with various edge supports. Benchmark solutions are obtained and analyzed in detail. Experimental results validate the convergence, stability, accuracy and reliability of the proposed approach. In particular, with a reasonable number of grid points, the new DSC‐Ritz method is capable of producing highly accurate numerical results for high‐mode vibration frequencies, which are hitherto unavailable to engineers. Moreover, the capability of predicting high modes endows us the privilege to reveal a discrepancy between natural higher‐order vibration modes of a Mindlin plate and those calculated via an analytical relationship linking Kirchhoff and Mindlin plates. Copyright © 2004 John Wiley & Sons, Ltd.

The mosaic of high-performance domain decomposition methods for structural mechanics––Part II: Formulation enhancements, multiple right-hand sides and implicit dynamics

Domain decomposition methods (DDM) constitute a family of high-performance solvers for large-scale problems of computational mechanics and of structural mechanics in particular. This paper compares popular DDM for structural analysis in both theory and computational performance, while it investigates the impact of enhanced mapping operators in their convergence behavior. Multiple right-hand side problems are considered, while standard and new DDM solution techniques for implicit dynamics are tested. The performed benchmark tests lead to useful conclusions regarding the comparative performance of the methods.

Assessment and applications of point interpolation methods for computational mechanics

AbstractTwo meshfree point interpolation methods (PIMs), which are based on the polynomial and the radial basis functions, have been proposed recently in addition to the earlier work with the moving least‐squares (MLS) approximation for the field function approximation. However, it is found that PIMs cannot automatically ensure the compatibility of the solution when they are used together with the energy principles. In this paper, issues related to the compatibility of PIMs are studied. A technique of background cell‐based nodal selections and a penalty method are proposed to enforce the compatibility of the solution of PIMs. The patch test is studied in great detail. The convergences and performances are investigated for both conforming and non‐conforming PIMs. It is found that those methods of the PIM family are very easy to implement, and are very efficient in obtaining numerical solutions for problems of computational mechanics. Copyright © 2004 John Wiley & Sons, Ltd.

Subdomain cluster generation for domain decomposition methods using graph partitioning optimization

Balancing and dual domain decomposition methods (DDMs) comprise a family of efficient high performance solution approaches for a large number of problems in computational mechanics. Such DDMs are used in practice on parallel computing environments with the number of generated subdomains being generally larger than the number of available processors. This paper presents an effective heuristic technique for organizing the subdomains into subdomain clusters, in order to assign each cluster to a processor. This task is handled by the proposed approach as a graph partitioning optimization problem using the publicly available software METIS. The objective of the optimization process is to minimize the communication requirements of the DDMs under the constraint of producing balanced processor workloads. This constraint optimization procedure for treating the subdomain cluster generation task leads to increased computational efficiencies for balancing and dual DDMs.

Modeling quasi-static crack growth with the extended finite element method Part II: Numerical applications

In Part I [Int. J. Solids Struct., 2003], we described the implementation of the extended finite element method (X-FEM) within Dynaflow™, a standard finite element package. In our implementation, we focused on two-dimensional crack modeling in linear elasticity. For crack modeling in the X-FEM, a discontinuous function and the near-tip asymptotic functions are added to the finite element approximation using the framework of partition of unity. This permits the crack to be represented without explicitly meshing the crack surfaces and crack propagation simulations can be carried out without the need for any remeshing. In this paper, we present numerical solutions for the stress intensity factor for crack problems, and also conduct crack growth simulations with the X-FEM. Numerical examples are presented with a two-fold objective: first to show the efficacy of the X-FEM implementation in Dynaflow™; and second to demonstrate the accuracy and versatility of the method to solve challenging problems in computational failure mechanics.

Generalized mixed variational principles and solutions of ill-conditioned problems in computational mechanics. Part II: Shear locking

Although the finite element method (FEM) has been extensively applied to various areas of engineering, the ill-conditioned problems occurring in many situations are still thorny to deal with. This study attempts to provide a high-performing and simple approach to the solutions of ill-conditioned problems. The theoretical foundation of it is the parametrized variational principles, called the generalized mixed variational principles (GMVPs) initiated by Rong in 1981. GMVPs can solve many kinds of ill-conditioned problems in computational mechanics. Among them, four cases are investigated in detail: the volumetric locking, the shear locking, the inhomogeneousness and the membrane locking problems, composing four parts of the study, Part I–Part IV, respectively. This paper is Part II, wherein a GMVP specially suited to the Reissner plate theory and Timoshenko beam theory is constructed, providing a mathematical foundation for establishing FEM formulations which can automatically unlock the shear locking and produce no spurious zero-energy modes.

Boundary meshfree methods based on the boundary point interpolation methods

A group of meshfree methods based on Boundary Integral Equation have been developed in order to overcome drawbacks in the conversional boundary element method (BEM) that requires boundary elements in constructing shape functions. In this paper, the radial basis point interpolation is firstly used to formulate the boundary radial point interpolation method (BRPIM). Two boundary meshfree methods: the boundary point interpolation method (BPIM) using the polynomial PIM and the BRPIM, are then examined in detail. The numerical implementations of these two methods are studied to address several technical issues, including the size of the compact support domain, the convergence, the performance, and so on. These two boundary-type meshfree methods are also compared with the Boundary Node Method and the conventional BEM in terms of both efficiency and performance. Several numerical examples of 2D elastostatics are analyzed using BPIM and BRPIM. It is found that BPIM and BRPIM are very easy to implement, and very robust for obtaining numerical solutions for problems of computational mechanics with very good accuracy. Key issues related the future development of boundary meshfree methods are also discussed.

A matrix triangularization algorithm for the polynomial point interpolation method

A novel matrix triangularization algorithm (MTA) is proposed to overcome the singularity problem in the point interpolation method (PIM) using the polynomial basis, and to ensure stable and reliable construction of PIM shape functions. The present algorithm is validated using several examples, and implemented in the local point interpolation method (LPIM) that is a truly meshfree method based on a local weak form. Numerical examples demonstrate that LPIM using the present MTA are very easy to implement, and very robust for solving problems of computational mechanics. It is shown that PIM with the present MTA is very effective in constructing shape functions. Most importantly, PIM shape functions possess Kronecker delta function properties. Parameters that influence the performance of them are studied in detail. The convergence and efficiency of them are thoroughly investigated.

Hybrid boundary point interpolation methods and their coupling with the element free Galerkin method

Hybrid boundary point interpolation methods (HBPIM and HBRPIM) are presented for solving boundary value problems of two-dimensional solids. In HBPIM and HBRPIM, the boundary of a problem domain is represented by properly scattered nodes. The point interpolation methods are used to construct shape functions with Kronecker delta function properties based on arbitrary distributed boundary nodes. Boundary conditions can be implemented with ease as in the conventional boundary element method. In HBPIM and HBRPIM, the ‘stiffness’ matrices so obtained are symmetric. This property of symmetry can be an added advantage in coupling the HBPIM and HBRPIM with other established meshfree methods. A novel coupled element free Galerkin (EFG)/HBPIM (or HBRPIM) method for 2D solids is then developed. The compatibility condition on the interface boundary is introduced into the variational formulations of HBPIM, HBRPIM and EFG using the Lagrange multiplier method. Coupled system equations are derived based on the variational formulation. The validity and efficiency of the present HBPIM, HBRPIM and coupled methods are demonstrated through the numerical examples. It is found that presented methods are very efficient for solving problems of computational mechanics.

Enhancing the performance of the FETI method with preconditioning ¶techniques implemented on clusters of networked computers

The FETI domain decomposition method for solving large-scale problems in computational structural mechanics involves the solution of an interface problem, which is handled by a Preconditioned Conjugate Projected Gradient (PCPG) algorithm. Two preconditioners are widely used to accelerate the convergence of the iterative PCPG algorithm: the optimal Dirichlet preconditioner and the economical lumped preconditioner. The Dirichlet preconditioner is computationally more efficient than the lumped preconditioner for ill-conditioned problems, but needs additional storage for the stiffness matrices of the subdomains' internal degrees of freedom (d.o.f.). In this study a new set of PCPG preconditioners is presented by providing approximate expressions to the inverse iteration matrix of the PCPG algorithm. The resulting approximate Dirichlet preconditioners are obtained by using instead of the whole stiffness matrix of the internal d.o.f. in each subdomain the following alternatives: a diagonal scaling matrix, a SSOR type matrix or an incomplete Cholesky factorization matrix. The computational behavior and performance of the proposed PCPG preconditioners is evaluated using an implementation of the FETI method on a cluster of ethernet-networked PCs running the message passing software PVM. It is demonstrated that the FETI method equipped with the approximate Dirichlet preconditioners leads for a number of large-scale problems to faster and less storage demanding overall solutions than with either Dirichlet or lumped preconditioner.

A vertex‐based finite volume method applied to non‐linear material problems in computational solid mechanics

AbstractA vertex‐based finite volume (FV) method is presented for the computational solution of quasi‐static solid mechanics problems involving material non‐linearity and infinitesimal strains. The problems are analysed numerically with fully unstructured meshes that consist of a variety of two‐ and three‐dimensional element types. A detailed comparison between the vertex‐based FV and the standard Galerkin FE methods is provided with regard to discretization, solution accuracy and computational efficiency. For some problem classes a direct equivalence of the two methods is demonstrated, both theoretically and numerically. However, for other problems some interesting advantages and disadvantages of the FV formulation over the Galerkin FE method are highlighted. Copyright © 2002 John Wiley & Sons, Ltd.

Comparisons of two meshfree local point interpolation methods for structural analyses

As truly meshless methods, the local point interpolation method (LPIM) and the local radial point interpolation method (LR-PIM), are based on the point interpolations and local weak forms integrated in a local domain of very simple shape. LPIM and LR-PIM are examined and compared with each other. They are also compared with the established FEM and the meshless local Petrov-Galerkin (MLPG) method. The numerical implementations of these two methods are discussed in detail. Parameters that influence the performance of them are detailedly studied. The convergence and efficiency of them are thoroughly investigated. LPIM and LR-PIM formulations are developed for structural analyses of 2-D elasto-dynamic problems and 1-D Timoshenko beam problems in the first time. It is found that LPIM and LR-PIM are very easy to implement, and very efficient obtaining numerical solutions to problems of computational mechanics.