Nonconforming decomposition methods: Techniques for linear elasticity

Mortar finite element methods provide a powerful tool for the numerical approximation of partial differential equations. Many domain decomposition techniques based on the coupling of different discretization schemes or of nonmatching triangulations along interior interfaces can be analyzed within this framework. Here, we present a mortar formulation based on dual basis functions and a special multigrid method. The starting point for our multigrid method is a symmetric positive definite system on the unconstrained product space. In addition, we introduce a new algorithm for the numerical solution of a nonlinear contact problem between two linear elastic bodies It will be shown that our method can be interpreted as an inexact Dirichlet-Neumann algorithm for the nonlinear problem. The boundary data transfer at the contact zone is essential for the algorithm. It is realized by a scaled mass matrix which results from a mortar discretization on non-matching triangulations with dual basis Lagrange multipliers. Numerical results illustrate the performance of our approach in 2D and 3D

Domain decomposition methods on nonmatching grids and some applications to linear elasticity problems

Domain decomposition techniques provide a powerful tool for the coupling of different discretization methods or nonmatching triangulations across subregion boundaries. Here, we consider mortar finite elements methods for linear elasticity and diffusion problems. These domain decomposition techniques provide a more flesible approach than standard conforming formulations. The mortar solution is weakly continuous at subregion boundaries, and its jump is orthogonal to a suitable Lagrange multiplier space. Our approach is based on dual bases for the Lagrange true for the standard mortar method [2]. The biorthogonality relation guarantees that the Lagrange multiplier can be locally eliminated, and that we obtain a symmetric positive semidefinite system on the unconstrained product space. This system will be solved by multigrid techniques. Numerical results illustrate the performance of the multigrid method in 2D and 3D

Monotone methods on non-matching grids for non-linear contact problems

Nonconforming domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. We use a generalized mortar method based on dual Lagrange multipliers for the discretization of a nonlinear contact problem between linear elastic bodies. In the case of unilateral contact problems, pointwise constraints occur and monotone multigrid methods yield efficient iterative solvers. Here, we generalize these techniques to nonmatching triangulations, where the constraints are realized in terms of weak integral conditions. The basic new idea is the construction of a nested sequence of nonconforming constrained spaces. We use suitable basis transformations and a multiplicative correction. In contrast to other approaches, no outer iteration scheme is required. The resulting monotone method is of optimal complexity and can be implemented as a multigrid method. Numerical results illustrate the performance of our approach in two and three dimensions

The surrogate matrix methodology: a priori error estimation

We give the first mathematically rigorous analysis of an emerging approach to finite element analysis (see, e.g., Bauer et al. [Appl. Numer. Math., 2017]), which we hereby refer to as the surrogate matrix methodology. This methodology is based on the piece-wise smooth approximation of the matrices involved in a standard finite element discretization. In particular, it relies on the projection of smooth so-called stencil functions onto high-order polynomial subspaces. The performance advantage of the surrogate matrix methodology is seen in constructions where each stencil function uniquely determines the values of a significant collection of matrix entries. Such constructions are shown to be widely achievable through the use of locally-structured meshes. Therefore, this methodology can be applied to a wide variety of physically meaningful problems, including nonlinear problems and problems with curvilinear geometries. Rigorous a priori error analysis certifies the convergence of a novel surrogate method for the variable coefficient Poisson equation. The flexibility of the methodology is also demonstrated through the construction of novel methods for linear elasticity and nonlinear diffusion problems. In numerous numerical experiments, we demonstrate the efficacy of these new methods in a matrix-free environment with geometric multigrid solvers. In our experiments, up to a twenty-fold decrease in computation time is witnessed over the classical method with an otherwise identical implementation

The surrogate matrix methodology: Low-cost assembly for isogeometric analysis

A new methodology in isogeometric analysis (IGA) is presented. This methodology delivers low-cost variable-scale approximations (surrogates) of the matrices which IGA conventionally requires to be computed from element-scale quadrature formulas. To generate surrogate matrices, quadrature must only be performed on certain elements in the computational domain. This, in turn, determines only a subset of the entries in the final matrix.The remaining matrix entries are computed by a simple B-spline interpolation procedure. Poisson’s equation, membrane vibration, plate bending, and Stokes’ flow problems are studied. In these problems, the use of surrogate matrices has a negligible impact on solution accuracy. Because only a small fraction of the original quadrature must be performed, we are able to report beyond a fifty-fold reduction in overall assembly time in the same software. The capacity for even further speed-ups is clearly demonstrated. The implementation used here was achieved by a small number of modifications to the open-source IGA software library GeoPDEs. Similar modifications could be made to other present-day software libraries

The surrogate matrix methodology: A reference implementation for low-cost assembly in isogeometric analysis

A reference implementation of a new method in isogeometric analysis (IGA) is presented. It delivers lowcost variable-scale approximations (surrogates) of the matrices which IGA conventionally requires to be computed by element-scale quadrature. To generate surrogate matrices, quadrature must only be performed on a fraction of the elements in the computational domain. In this way, quadrature determines only a subset of the entries in the final matrix. The remaining matrix entries are computed by a simple B-spline interpolation procedure. We present the modifications and extensions required for a reference implementation in the open-source IGA software library GeoPDEs. The exposition is fashioned to help facilitate similar modifications in other contemporary software libraries. Method name: Surrogate matrix method for isogeometric analysi

Scheduling Massively Parallel Multigrid for Multilevel Monte Carlo Methods

The computational complexity of naive, sampling-based uncertainty quantification for 3D partial differential equations is extremely high. Multilevel approaches, such as multilevel Monte Carlo (MLMC), can reduce the complexity significantly when they are combined with a fast multigrid solver, but to exploit them fully in a parallel environment, sophisticated scheduling strategies are needed. We optimize the concurrent execution across the three layers of the MLMC method: parallelization across levels, across samples, and across the spatial grid. In a series of numerical tests, the influence on the overall performance of the “scalability window” of the multigrid solver (i.e., the range of processor numbers over which good parallel efficiency can be maintained) is illustrated. Different homogeneous and heterogeneous scheduling strategies are proposed and discussed. Finally, large 3D scaling experiments are carried out, including adaptivity

Biorthogonal splines for optimal weak patch-coupling in isogeometric analysis with applications to finite deformation elasticity

A new construction of biorthogonal splines for isogeometric mortar methods is proposed. The biorthogonal basis has a local support and, at the same time, optimal approximation properties, which yield optimal results with mortar methods. We first present the univariate construction, which has an inherent crosspoint modification. The multivariate construction is then based on a tensor product for weighted integrals, whereby the important properties are inherited from the univariate case. Numerical results including large deformations confirm the optimality of the newly constructed biorthogonal basis.Comment: biorthogonal basis, finite deformation, isogeometric analysis, mortar methods, multi-patch geometrie