Lothar Banz

On h p -adaptive BEM for frictional contact problems in linear elasticity

A mixed formulation for a Tresca frictional contact problem in linear elasticity is considered in the context of boundary integral equations, which is later extended to Coulomb friction. The discrete Lagrange multiplier, an approximation of the surface traction on the contact boundary part, is a linear combination of biorthogonal basis functions. In case of curved (isoparametric) elements, these are the solutions of local problems. In particular, the biorthogonality allows to rewrite the variational inequality constraints as a simple set of complementarity problems. Thus, enabling an efficient application of a semi-smooth Newton solver for the discrete mixed problem, converging locally super-linearly in the frictional case and quadratically in the frictionless case. Typically, the solution of frictional contact problems is of reduced regularity at the interface between contact to non-contact and from stick to slip. To identify the a priori unknown locations of these interfaces two a posteriori error estimations are introduced. In a first step the error is split into specific error contributions resulting from the contact and friction conditions and from the discretization error of a variational equation. For the latter a residual and a bubble error estimation are considered explicitly. The numerical experiments show the applicability of the derived error estimations and the superiority of h p -adaptivity compared to low order uniform and adaptive approaches.

hp-adaptive IPDG/TDG-FEM for parabolic obstacle problems

For a parabolic obstacle problem two equivalent hp-FEM discretization methods based on interior penalty discontinuous Galerkin in space and discontinuous Galerkin in time are presented. The first approach is based on a variational inequality (VI) formulation and the second approach on a mixed method in which the non-penetration condition is resolved by a Lagrange multiplier. The discrete Lagrange multiplier is a linear combination of biorthogonal basis functions, allowing to write the discrete VI-constraints as a set of complementarity problems. Employing a penalized Fischer-Burmeister non-linear complementarity function, the discrete mixed problem can be solved by a locally Q-quadratic converging semi-smooth Newton (SSN) method. The hierarchical a posteriori error estimator for the VI-formulation, which under the saturation assumption is both efficient and reliable, allows hp-adaptivity. The numerical experiments show improved convergence compared to uniform and h-adaptive meshes. Furthermore, an a priori error estimate is given for the VI-formulation.

Stabilized mixed hp-BEM for frictional contact problems in linear elasticity

We analyze stabilized mixed hp-boundary element methods for frictional contact problems for the Lamé equation. The stabilization technique circumvents the discrete inf-sup condition for the mixed problem and thus allows us to use the same mesh and polynomial degree for the primal and dual variables. We prove a priori convergence rates in the case of Tresca friction, using Gauss-Legendre-Lagrange polynomials as test and trial functions for the Lagrange multiplier. Additionally, a residual based a posteriori error estimate for a more general class of discretizations is derived. It in particular applies to discretizations based on Bernstein polynomials for the discrete Lagrange multiplier, which we also analyze. The discretization and the a posteriori error estimate are extended to the case of Coulomb friction. Several numerical experiments underline our theoretical results, demonstrate the behavior of the method and its insensitivity to the scaling and perturbations of the stabilization term.

Comparison of mixed h p -BEM (stabilized and non-stabilized) for frictional contact problems

We present boundary integral equation procedures for contact problems (with Tresca or Coulomb friction) which are based on mixed formulations where besides the displacement also the traction on the contact boundary part appears (as a Lagrange multiplier). This approach allows for an easy and efficient way to perform an h p -BE method by the use of biorthogonal basis functions. This is especially suited for applying the semi-smooth Newton (SSN) method which is a very efficient solver superior to standard algorithm like Uzawa. With an adaptive algorithm we perform locally mesh refinements and increase of polynomial degrees for the BE solution-thus correctly representing the contact phenomena. We also present as stabilized version of our mixed h p -BEM scheme with Gauss-Lobatto-Lagrange basis which circumvents the discrete inf-sup condition. Numerical results supporting our theory are reported.

Higher order FEM for the obstacle problem of the p-Laplacian—A variational inequality approach

Abstract We consider higher order finite element discretizations of a nonlinear variational inequality formulation arising from an obstacle problem with the p -Laplacian differential operator for p ∈ ( 1 , ∞ ) . We prove an a priori error estimate and convergence rates with respect to the mesh size h and in the polynomial degree q under assumed regularity. Moreover, we derive a general a posteriori error estimate which is valid for any uniformly bounded sequence of finite element functions. All our results contain the known results for the linear case of p = 2 . We present numerical results on the improved convergence rates of adaptive schemes (mesh size adaptivity with and without polynomial degree adaptation) for the singular case of p = 1 . 5 and for the degenerated case of p = 3 .

High precision modeling towards the 10-20 level†

The requirements for accurate numerical simulations are increasing steadily. Modern high precision physics experiments now exceed the achievable numerical accuracy of standard commercial and scientific simulation tools. One example are optical resonators for which changes in the optical length are now commonly measured to 10-15 precision. The achievable measurement accuracy for resonators and cavities is directly influenced by changes in the distances between the optical components. If deformations in the range of 10-15 occur, those effects cannot be modeled and analyzed anymore with standard methods based on double precision data types. New experimental approaches point out that the achievable experimental accuracies may improve up to the level of 10-17 in the near future. For the development and improvement of high precision resonators and the analysis of experimental data, new methods have to be developed which enable the needed level of simulation accuracy. Therefore we plan the development of new high precision algorithms for the simulation and modeling of thermo-mechanical effects with an achievable accuracy of 10-20. In this paper we analyze a test case and identify the problems on the way to this goal.