Finite-element Solution Of Problems Research Articles

The inf-sup stability of the lowest order Taylor–Hood pair on affine anisotropic meshes

Abstract Uniform inf-sup conditions are of fundamental importance for the finite element solution of problems in incompressible fluid mechanics, such as the Stokes and Navier–Stokes equations. In this work we prove a uniform inf-sup condition for the lowest-order Taylor–Hood pairs $\mathbb{Q}_2\times \mathbb{Q}_1$ and $\mathbb{P}_2\times \mathbb{P}_1$ on a family of affine anisotropic meshes. These meshes may contain refined edge and corner patches. We identify necessary hypotheses for edge patches to allow uniform stability and sufficient conditions for corner patches. For the proof, we generalize Verfürth’s trick and recent results by some of the authors. Numerical evidence confirms the theoretical results.

ИСПОЛЬЗОВАНИЕ ПРЕПРОЦЕССОРА FEMAP СОВМЕСТНО С КОМБИНИРОВАННЫМ МЕТОДОМ РЕШЕНИЯ НЕЛИНЕЙНЫХ КОНЕЧНОЭЛЕМЕНТНЫХ ЗАДАЧ

Earlier in BSTU there was developed a universal but comparatively complex combined algorithm for the solution of finite element problems in which resolving equations can take into account different nonlinearity. An attempt is made to use a modern interface of the FEMAP program package jointly with the program realizing this algorithm. In fact a convertor of data prepared in FEMAP formats required for the original program is developed. A new program formed is tested by some examples. Now the program realizing a combined algorithm has a modern interface and it is convenient for use.

Efficient micromechanical analysis of fiber-reinforced composites subjected to cyclic loading through time homogenization and reduced-order modeling

In this paper, a number of techniques used to accelerate the solution of finite element problems involving a large number of load cycles areexplored and applied to the micromechanical analysis of fiber-reinforced composites. The microscopic domain consists of unidirectional linear-elastic fibers embedded in a viscoelastic/viscoplastic polymeric matrix. Time homogenization is applied to divide the original equilibrium problem in macro- and microchronological parts. The size of the problem is further reduced by a combination of Proper Orthogonal Decomposition (POD) and the Empirical Cubature Method (ECM), resulting in a hyper-reduced model. A novel technique for history recovery combining Gappy Data reconstruction with a k-means clustering algorithm is proposed, as well as an adaptive strategy combining time homogenization and POD without an offline training phase. The performance of each acceleration technique is assessed and the resultant speed-ups obtained by combining them are presented.

Treatment of Block-Based Sparse Matrices in Domain Decomposition Method

Abstract— The domain decomposition method involves the finite element solution of problems in the parallel computer. The finite element discretization leads to the solution of large systems of linear equation whose matrix is naturally sparse. The use of proper storing techniques for sparse matrix is fundamental especially when dealing with large scale problems typical of industrial applications. The aim of this research is to review the sparsity pattern of the matrices originating from the discretization of the elasto-plastic and thermal-convection problems. Some practical strategies dealing with sparsity pattern in the finite element code of adventure system are recalled. Several efficient storage schemes to store the matrix originating from elasto-plastic and thermal-convection problems have been proposed. In the proposed technique, inherent block pattern of the matrix is exploited to locate the matrix element. The computation in the high performance computer shows better performance compared to the conventional skyline storage method used by the most of the researchers.

An adaptive strategy for the bivariate solution of finite element problems using multivariate nested Padé approximants

SUMMARYMost engineering applications involving solutions by numerical methods are dependent on several parameters, whose impact on the solution may significantly vary from one to the other. At times an evaluation of these multivariate solutions may be required at the expense of a prohibitively high computational cost. In the present paper, an adaptive approach is proposed as a way to estimate the solution of such multivariate finite element problems. It is based upon the integration of so‐called nested Padé approximants within the finite element procedure. This procedure includes an effective control of the approximation error, which enables adaptive refinements of the converged intervals upon reconstruction of the solution. The main advantages lie in a potential reduction of the computational effort and the fact that the level of a priori knowledge required about the solution in order to have an accurate, efficient, and well‐sampled estimate of the solution is low. The approach is introduced for bivariate problems, for which it is validated on elasto‐poro‐acoustic problems of both academic and more industrial scale. It is argued that the methodology in general holds for more than two variables, and a discussion is opened about the truncation refinements required in order to generalize the results accordingly. Copyright © 2014 John Wiley & Sons, Ltd.

Solution of finite element problems using hybrid parallelization with MPI and OpenMP

The Finite Element Method (FEM) is used to solve problems like solid deformation and heat diffusion in domains with complex geometries. This kind of geometries requires discretization with millions of elements; this is equivalent to solve systems of equations with sparse matrices and tens or hundreds of millions of variables. The aim is to use computer clusters to solve these systems. The solution method used is Schur substructuration. Using it is possible to divide a large system of equations into many small ones to solve them more efficiently. This method allows parallelization. MPI (Message Passing Interface) is used to distribute the systems of equations to solve each one in a computer of a cluster. Each system of equations is solved using a solver implemented to use OpenMP as a local parallelization method.

Robustness of error estimators for finite element solutions of problems with high orthotropy

In this paper, we address the accuracy of a posteriori error estimators for finite element solutions of problems with high orthotropy especially for cases where rather coarse meshes are used, which are often encountered in engineering computations. We present sample computations which indicate lack of robustness of all standard residual estimators with respect to high orthotropy. The main culprit is the coarseness of the finite element meshes relative to the thickness of the boundary and interface layers in the solution. This is often imposed by the size of the problem domain, the capabilities for mesh generation, and the employed finite element software and cannot be changed. Hence, there is need for a posteriori error estimation approaches which are robust for coarse meshes and which are constructed using non-intrusive approaches with respect to the employed software.

Parallel methods for optimality criteria-based topology optimization

Topology optimization problems require the repeated solution of finite element problems that are often extremely ill-conditioned due to highly heterogeneous material distributions. This makes the use of iterative linear solvers inefficient unless appropriate preconditioning is used. Even then, the solution time for topology optimization problems is typically very high. These problems are addressed by considering the use of non-overlapping domain decomposition-based parallel methods for the solution of topology optimization problems. The parallel algorithms presented here are based on the solid isotropic material with penalization (SIMP) formulation of the topology optimization problem and use the optimality criteria method for iterative optimization. We consider three parallel linear solvers to solve the equilibrium problem at each step of the iterative optimization procedure. These include two preconditioned conjugate gradient (PCG) methods: one using a diagonal preconditioner and one using an incomplete LU factorization preconditioner with a drop tolerance. A third substructuring solver that employs a hybrid of direct and iterative (PCG) techniques is also studied. This solver is found to be the most effective of the three solvers studied, both in terms of parallel efficiency and in terms of its ability to mitigate the effects of ill-conditioning. In addition to examining parallel linear solvers, we consider the parallelization of the iterative optimality criteria method. To tackle checkerboarding and mesh dependence, we propose a multi-pass filtering technique that limits the number of “ghost” elements that need to be exchanged across interprocessor boundaries.

A novel approach to circuit-field-torque coupled time stepping finite element modeling of electric machines

This paper presents a sub-block algorithm for the time stepping finite element solution of problems in which sets of electromagnetic field equations, circuit equations and mechanical equation are coupled together. The proposed method ensures that identical solutions are obtained for the sub-blocks by controlling the step size of the time stepping process. This new method simplifies the process of dealing with coupled-system problems and it also greatly reduces the computation time. A time stepping finite element model of an induction motor is used to demonstrate the proposed method in details.

A fuzzy arithmetical approach to the solution of finite element problems with uncertain parameters

A fuzzy arithmetical approach to the solution of finite element problems with uncertain parameters

Non-Reflecting Finite Element Schemes for Three-Dimensional Acoustic Waves

The finite element solution of problems involving three-dimensional acoustic waves in an infinite wave guide, and in the infinite medium around a structure is considered. Such problems are typical in structural acoustics, and this paper concentrates on the efficient numerical treatment of the infinite acoustic medium away from the structure. The unbounded domain is truncated by means of an artificial boundary ℬ. On ℬ, non-reflecting boundary conditions are used; these are either nonlocal Dirichlet-to-Neumann conditions, or their localized counterparts. For the high-order localized conditions, special three-dimensional finite elements are constructed for use in the layer adjacent to ℬ. The performance of the nonlocal and localized boundary conditions is compared via numerical experiments involving a three-dimensional wave guide.

Evaluation of energy release rate for a planar crack in heterogeneous media

A modified version of domain integral method is developed for evaluation of energy release rate with finite element solutions for problems with a 2-D crack located in a heterogeneous elastic field. The heterogeneous field considered in this work generally contains various materials, with discontinuous mechanical moduli across the interfaces. The formulation is proved to be patch-independent, in a generalized sense, and valid for problems under both small and large deformations. The results of calculation appear to be very insensitive to the crack tip finite element models when the tip is away from the material interface. However, strong dependency on the local modeling is observed in case the tip is located at the interface. Alternative studies on this particular case are thus required.

Parallel algorithms for maximal monotone operators of local type

This paper presents general algorithms for the parallel solution of finite element problems associated with maximal monotone operators of local type. The latter concept, which is also introduced here, is well suited to capture the idea that the given operator is the discretization of a differential operator that may involve nonlinearities and/or constraints as long as those are of a local nature. Our algorithms are obtained as a combination of known algorithms for possibly multi-valued maximal monotone operators with appropriate decompositions of the domain. This work extends a method due to two of the authors in the single-valued and linear case.

The inf-sup test

We briefly review the inf-sup condition for the finite element solution of problems in incompressible elasticity, and then propose a numerical test on whether the inf-sup condition is passed. The evaluation of elements with this test is simple, and various results are presented. This inf-sup test will prove useful for many discretizations of constrained variational problems.

Application of mathematica to the direct semi-numerical solution of finite element problems

Several numerical methods, such as the finite element method, reduce applied mechanics and additional engineering problems to systems of linear algebraic equations. It has been already suggested that the inclusion of a symbolic parameter in the corresponding numerical results leads to a generality and a wide applicability of these results. Here we suggest the direct solution of these equations by using the popular computer algebra system MATHEMATICA. Assuming the results expressed in a Taylor-Maclaurin series form with respect to the selected symbolic parameter, the whole problem is reduced to the solution of an appropriate number of systems of purely numerical linear equations. This can be achieved either inside MATHEMATICA or by using efficient external numerical routines. As an application the above modification of the finite element method was used in the classical problem of a tapered elastic beam. The obtained semi-numerical results by the finite element method were seen to be in agreement with the available theoretical results. Further possibilities are also suggested in brief.

Treatment of internal constraints by mixed finite element methods: Unification of concepts

AbstractA general method to generate assumed stress and strain fields within the context of mixed finite element methods is presented. The assumed fields are constructed in such a way that internal constraints are satisfied a priori. Consequently, the locking behaviour commonly observed in finite element solutions of problems with internal constraints is avoided. To this end, the assumed stress and strain fields are constructed to satisfy a priori the homogeneous part of the equilibrium equations, thus avoiding Fraeijs de Veubeke's limitation principle. Results obtained using the proposed methodology on a nearly incompressible plane strain problem and thin plate application using a shear deformable theory are indicated.

The spectral overlay on finite elements for problems with high gradients

A methodology has been developed for enhancing the accuracy of finite element solutions of problems with high gradients. This is accomplished by superimposing the spectral approximation on subdomains which overlay the finite element mesh in regions where high gradients are indicated by the solution. The orientation and shape of these subdomains are quite arbitrary. Results are obtained for a viscoplastic bar which is stretched beyond its bifurcation point. The results show that with a modest increase in the number of spectral unknowns, it is possible to achieve high resolution of the strain field in the shear band.

A procedure for a posteriori error analysis for the finite element method which contains a bounding measure

Post-processing procedures are applied to estimate the error in energy, displacement and stress in finite element solutions for problems in linear elasticity. The error estimates are based on a complementary theory which can ensure that the estimate of the energy of the error is an upper bound. Local estimates of the error in displacement and stress follow either directly from the complementary analysis, or by extrapolating the solution to the energy bound.

On the mapping of massively parallel processors onto finite element graphs

Massively parallel processors have been built and some are commercially available. Efficient solution of finite element problems on these machines requires minimizing interprocessor communication bandwidth. This is achieved by mapping adjacent generalized elements as far as possible onto directly connected processors. When the topology of the multiprocessor embeds the communication structure of the finite element mesh, this mapping problem becomes similar to the problem of matching two graphs. However, realistic finite element meshes which include several discretization patterns and a variety of local refinement procedures do not result in graphs that exactly match the wiring of a parallel processor. For this reason, we concentrate here on efficient heuristics that produce a mapping that can minimize the overall communication cost and/or communication bandwidth. Two algorithms are presented, one for MIMD machines and another for SIMD computers. It is shown that over a trivial mapping and with a quite reasonable complexity, both algorithms can significantly reduce the cost/bandwidth of communication of a massively parallel processor.

Finite element solutions of waveguide-interconnection problems

The coupling of waveguides is modeled to a mathematical problem for an internal region. The transition region may have a spatially dependent refractive index and irregular boundaries. Finite element solutions of this problem extending into the longitudinal direction are formulated. The resulting indefinite, non symmetric and sparse set of algebraic equations can be solved by an efficient algorithm.