Advanced Finite Element Methods and Applications

The paper recalls the period 1988-1993 when the research on parallel algorithms and their implementation started in Karl-Marx-Stadt (renamed to Chemnitz in 1990).We consider the research group formed at this time and the hardware available to this group. Parallel hardware as the transputer is considered and the ancient parallel computers from that time are depicted. The group has been formed by the series of workshops and seminars that took place; and the FEM-Symposium is still organized annually. We will focus on a few of these activities and present the developments in hardware, numerical methods, parallel algorithms and analysis that have been discussed between professors, research assistants and students. The paper contains also a brief view on parallel computers available to that group today and some examples document how the computing power has increased during a period of more than 20 years.

Second order elliptic equations are considered in the unit square, which is decomposed into subdomains by an arbitrary nonuniform orthogonal grid. For the elliptic operator we assume that the energy integral contains only squares of first order derivatives with coefficients, which are arbitrary positive finite numbers but different for each subdomain. The orthogonal finite element mesh has to satisfy only one condition: it is uniform on each subdomain. No other conditions on the coefficients of the elliptic equation and on the step sizes of the discretization and decomposition are imposed. For the resulting discrete finite element problem, we suggest domain decomposition algorithms of linear total arithmetical complexity, not depending on any of the three factors contributing to the orthotropism of the discretization on subdomains. The main problem of designing such an algorithm is the preconditioning of the inter-subdomain Schur complement, which is related in part to obtaining boundary norms for discrete harmonic functions on the shape irregular domains.

In this article, we provide a rigorous a priori error estimate for the symmetric coupling of the finite and boundary element method for the potential problem in three dimensions. Our theoretical framework allows an arbitrary number of polyhedral subdomains. Our bound is not only explicit in the mesh parameter, but also in the subdomains themselves: the bound is independent of the number of subdomains and involves only the shape regularity constants of a certain coarse triangulation aligned with the subdomain decomposition. The analysis includes the so-called BEM-based FEM as a limit case.

This review gives an overview of current anisotropic refinement methods in finite elements, with the focus on the actual refinement step. In this we highlight strengths and weaknesses of different approaches and hope to stimulate research into closer coupling of the refinement process with efficient solution strategies for the equation systems arising from the discretized equations. A rough overview of different categories of error estimation techniques relevant in the anisotropic setting is also given.

We present a new finite element algorithm for computing the stress intensity factors and the solution of boundary value problems for the Poisson equation in two-dimensional domains with corners. The method makes use of an explicit expression for the stress intensity factors in terms of the function of the right hand side, the solution of the boundary value problem and smooth cutoff functions to compute from an initial finite element solution approximations of the stress intensity factors. The computed values of the stress intensity factors are then used for post processing the finite element solution and the approximated stress intensity factors. The algorithm leads to good approximations of both stress intensity factors and the finite element solution of the boundary value problem on quasi regular meshes. The results are illustrated by numerical experiments.

In many areas of retail and especially e-business recommendation engines are applied to increase the usability of the store or portal. Advanced recommendation engines use approaches from control theory for adaptive learning. At the forefront of these algorithms reinforcement learning is applied which however requires large transaction numbers to converge. To overcome this problem, we propose a hierarchical approach of reinforcement learning for recommendation engines by combining a multilevel preconditioner with the temporal-difference learning method, the most important algorithm class of reinforcement learning. The multilevel preconditioner works on a combined hierarchy of states and actions. We describe the preconditioner, prove its convergence and present results on real-life data.

We review some of the recent work on preconditioners for saddle point problems. In particular, we discuss preconditioners that are constructed based on exact or inexact Schur complements and on interpolation theory. These preconditioners are used within Krylov subspace methods, for which it is shown that the total number of iterations is bounded by global constants. The described techniques are applied to two model problems from optimal control.

The convergence analysis of iterative methods for nonlinear eigenvalue problems is in the most cases restricted either to algebraically simple eigenvalues or to polynomial eigenvalue problems. In this paper we consider two classical methods for general holomorphic eigenvalue problems, namely the nonlinear generalized Rayleigh quotient iteration (NGRQI) and the augmented Newton method. The analysis of the convergence order of both methods is based on the representation of the eigenvalues as poles of the resolvent. This approach was already chosen for the analysis of the NGRQI by Langer in [19] for a more general setting where such a representation of the eigenvalues had to be assumed. The convergence orders of both methods depend on the order which an eigenvalue has as pole of the resolvent. Both methods exhibit a local quadratic convergence order for semi-simple eigenvalues. For defective eigenvalues in general only a local linear convergence is possible. In numerical experiments the theoretical results are confirmed.

In this paper we focus on the sensitivity analysis of Maxwell’s eigenvalue problem, where the derivatives of the eigenvalues are calculated with respect to design parameters (i.e., material or geometrical parameters). Utilizing the adjoint approach the derivatives can be calculated at almost no additional cost. The challenge consists in the computation of the required derivatives (i.e., derivatives of bilinear forms with respect to the design parameters) from a higher order, curved finite element discretization. Numerical studies show the application for a real life electromagnetic filter application where the sensitivities of the eigenvalues give a better insight into the characteristics of the underlying filter. The benefit is apparent if the adjoint method is compared to a standard finite difference approach.

Optical field tracing methods generalize ray tracing methods by considering harmonic fields instead of ray bundles. This allows the smooth combination of different modeling techniques in different subdomains of the system. Based on tearing and interconnecting ideas, the paper introduces the basic concepts of non-sequential field tracing and derives the corresponding operator equations and a solution formula for the simulation task. The evaluation requires the solution of local Maxwell problems (tearing) and the continuity of the solution across boundaries is achieved along with the convergence of the iterative procedure (interconnecting). The number of local problems to be solved is optimized by a newly introduced light path tree algorithm. Finally some examples for the selection of local Maxwell solvers and numerical results are presented.

A boundary integral formulation for a mixed boundary value problem in linear elastostatics with a conservative right hand side is considered. A meshless interpolant of the scalar potential of the volume force density is constructed by means of radial basis functions. An exact particular solution to the Lamé system with the gradient of this interpolant as the right hand side is found. Thus, the need of approximating the Newton potential is eliminated. The procedure is illustrated on numerical examples.

This contribution presents advanced numerical models for the solution of the direct and inverse problems of nearly incompressible hyperelastic processes at large strains. The discussed mixed finite element approach contributes to the numerical simulation of coupled multiphysics problems, including the calibration of appropriate material models (parameter identification). The presented constitutive approach is based on the multiplicative decomposition of the deformation gradient resulting in a two-field formulation with displacement components and hydrostatic pressure as primary variables. The ill-posed inverse problem of parameter identification analyzing inhomogeneous displacement fields is solved using deterministic trust-region optimization techniques.Within this context, a semi-analytical approach for sensitivity analysis represents an efficient and accurate method to determine the gradient of the objective function. Themixed boundary value problem is based on the spatial discretization of the weak formulations of the linear momentum balance and the incompressibility condition. Its linearization serves as basis for the solution of the direct problem, while the implicit differentiation of the weak formulations with respect to material parameters provides the necessary relations for the semi-analytical sensitivity analysis. Adaptive mesh refinement and mesh coarsening are realized controlled by a residual a posteriori error estimator. Efficiency and accuracy of the presented direct and inverse numerical techniques are demonstrated on a typical example.

This paper provides an extension of the reciprocity gap approach for crack detection from electrostatics [1], isotropic [2] and anisotropic linear elasticity [18] to piezoelectric materials. We show unique and stable identifiability of the crack plane fromone or two pairs of appropriate Dirichlet-Neumann data and illustrate the approach by numerical tests with simulated data obtained by adaptive finite element computations.

In this paper, a finite element software for automated simulation of fatigue crack growth in arbitrarily loaded three-dimensional components is presented. The criterion, direction and amount of crack propagation are controlled by concepts of linear elastic fracture mechanics. The fracture mechanical parameters are calculated by means of a special submodelling technique in combination with the interaction integral or the virtual crack closure technique. In the adaptive crack growth step, the updated crack front position is determined and the mesh in the crack region is automatically adapted. The preprocessing and main FEM-analysis of the cracked structure are done using the commercial software ABAQUS. Two application examples show the capability and performance of the simulation program.