arXiv subjects

Oliver Sander

Publications and source records attributed to Oliver Sander.

46 records · Page 3Linked to original sources

arXiv2017

Truncated Nonsmooth Newton Multigrid Methods for Block-Separable Minimization Problems

The Truncated Nonsmooth Newton Multigrid (TNNMG) method is a robust and efficient solution method for a wide range of block-separable convex minimization problems, typically stemming from discretizations of nonlinear and nonsmooth partial differential equations. This paper proves global convergence of the method under weak conditions both on the objective functional, and on the local inexact subproblem solvers that are part of the method. It also discusses a range of algorithmic choices that allows to customize the algorithm for many specific problems. Numerical examples are deliberately omitted, because many such examples have already been published elsewhere.

math.NA↗

arXiv2017

Solving primal plasticity increment problems in the time of a single predictor-corrector iteration

The Truncated Nonsmooth Newton Multigrid (TNNMG) method is a well-established method for the solution of strictly convex block-separably nondifferentiable minimization problems. It achieves multigrid-like performance even for non-smooth nonlinear problems, while at the same time being globally convergent and without employing penalty parameters. We show that the algorithm can be applied to the primal problem of classical linear elastoplasticity with hardening. Numerical experiments show that the method is considerably faster than classical predictor-corrector methods. Indeed, solving an entire increment problem with TNNMG takes less time than a single predictor-corrector iteration for the same problem. Since the algorithm does not rely on differentiability of the objective functional, nonsmooth yield functions like the Tresca yield function can be easily incorporated. The method is closely related to a predictor-corrector scheme with a consistent tangent predictor and line search. We explain the algorithm, prove global convergence, and show its efficiency using a standard benchmark from the literature.

math.NA↗

arXiv2017

A globally convergent filter-trust-region method for large deformation contact problems

We present a globally convergent method for the solution of frictionless large deformation contact problems for hyperelastic materials. The discretisation uses the mortar method which is known to be more stable than node-to-segment approaches. The resulting non-convex constrained minimisation problems are solved using a filter-trust-region scheme, and we prove global convergence towards first-order optimal points. The constrained Newton problems are solved robustly and efficiently using a Truncated Non-smooth Newton Multigrid (TNNMG) method with a Monotone Multigrid (MMG) linear correction step. For this we introduce a cheap basis transformation that decouples the contact constraints. Numerical experiments confirm the stability and efficiency of our approach.

math.NA↗

arXiv2017

Simulating Wear On Total Knee Replacements

Wear on total knee replacements (TKRs) is an important criterion for their performance characteristics. Numerical simulations of such wear has seen increasing attention over the last years. They have the potential to be much faster and less expensive than the in vitro tests in use today. While it is unlikely that in silico tests will replace actual physical tests in the foreseeable future, a judicious combination of both approaches can help making both implant design and pre-clinical testing quicker and more cost-effective. The challenge today for the design of simulation methods is to obtain results that convey quantitative information, and to do so quickly and reliably. This involves the choice of mathematical models as well as the numerical tools used to solve them. The correctness of the choice can only be validated by comparing with experimental results. In this paper we present finite element simulations of the wear in TKRs during the gait cycle standardized in the ISO 14243-1 document, used for compliance testing in several countries. As the ISO 14243-1 standard is precisely defined and publicly available, it can serve as an excellent benchmark for comparison of wear simulation methods. Our novel contact algorithm works without Lagrange multipliers and penalty methods, achieving unparalleled stability and efficiency. We compare our simulation results with the experimental data from physical tests using two different actual TKRs, each test being performed three times. We can closely predict the total mass loss due to wear after five million gait cycles. We also observe a good match between the wear patterns seen in experiments and our simulation results.

q-bio.TO↗

arXiv2016

Simulation of Deformation and Flow in Fractured, Poroelastic Materials

We introduce a coupled system of PDEs for the modeling of the fluid-fluid and fluid-solid interaction in a poroelastic material with a single static fracture. The fluid flow in the fracture is modeled by a lower-dimensional Darcy equation, which interacts with the surrounding rock matrix and the fluid it contains. We explicitly allow the fracture to end within the domain, and the fracture width is an unknown of the problem. The resulting weak problem is nonlinear, elliptic and symmetric, and can be given the structure of a fixed-point problem. We show that the coupled fluid-fluid problem has a solution in a specially crafted Sobolev space, even though the fracture width cannot be bounded away from zero near the crack tip. For numerical simulations, we combine XFEM discretizations for the rock matrix deformation and pore pressure with a standard lower-dimensional finite element method for the fracture flow problem. The resulting coupled discrete system consists of linear subdomain problems coupled by nonlinear coupling conditions. We solve the coupled system with a substructuring solver and observe very fast convergence. We also observe optimal mesh dependence of the discretization errors even in the presence of crack tips.

math.NA↗

arXiv2016

Test Function Spaces for Geometric Finite Elements

We construct test function spaces for geometric finite elements. Geometric finite elements (GFE) are generalizations of Lagrangian finite elements to situations where the unknown function maps into a nonlinear space. Test functions for such spaces arise as variations of GFE functions wherever the GFE function space has a local manifold structure. For any given GFE function $u_h$, the test functions form a linear space that depends on $u_h$. They generalize Jacobi fields in the same way that the GFE interpolation functions generalize geodesic curves. Having test function spaces allows to extend the GFE method to boundary value problems that do not have a minimization formulation.

math.NA↗

arXiv2015

The interface for functions in the dune-functions module

The dune-functions dune module introduces a new programmer interface for discrete and non-discrete functions. Unlike the previous interfaces considered in the existing dune modules, it is based on overloading operator(), and returning values by-value. This makes user code much more readable, and allows the incorporation of newer C++ features such as lambda expressions. Run-time polymorphism is implemented not by inheritance, but by type erasure, generalizing the ideas of the std::function class from the C++11 standard library. We describe the new interface, show its possibilities, and measure the performance impact of type erasure and return-by-value.

cs.MS↗

arXiv2015

The Dune FoamGrid implementation for surface and network grids

We present FoamGrid, a new implementation of the DUNE grid interface. FoamGrid implements one- and two-dimensional grids in a physical space of arbitrary dimension, which allows for grids for curved domains. Even more, the grids are not expected to have a manifold structure, i.e., more than two elements can share a common facet. This makes FoamGrid the grid data structure of choice for simulating structures such as foams, discrete fracture networks, or network flow problems. FoamGrid implements adaptive non-conforming refinement with element parametrizations. As an additional feature it allows removal and addition of elements in an existing grid, which makes FoamGrid suitable for network growth problems. We show how to use FoamGrid, with particular attention to the extensions of the grid interface needed to handle non-manifold topology and grid growth. Three numerical examples demonstrate the possibilities offered by FoamGrid.

cs.MS↗

arXiv2015

Numerical Treatment of a Geometrically Nonlinear Planar Cosserat Shell Model

We present a new way to discretize a geometrically nonlinear elastic planar Cosserat shell. The kinematical model is similar to the general 6-parameter resultant shell model with drilling rotations. The discretization uses geodesic finite elements, which leads to an objective discrete model which naturally allows arbitrarily large rotations. Finite elements of any approximation order can be constructed. The resulting algebraic problem is a minimization problem posed on a nonlinear finite-dimensional Riemannian manifold. We solve this problem using a Riemannian trust-region method, which is a generalization of Newton's method that converges globally without intermediate loading steps. We present the continuous model and the discretization, discuss the properties of the discrete model, and show several numerical examples, including wrinkles of thin elastic sheets in shear.

math.NA↗

arXiv2013

Unsaturated subsurface flow with surface water and nonlinear in- and outflow conditions

We analytically and numerically analyze groundwater flow in a homogeneous soil described by the Richards equation, coupled to surface water represented by a set of ordinary differential equations (ODE's) on parts of the domain boundary, and with nonlinear outflow conditions of Signorini's type. The coupling of the partial differential equation (PDE) and the ODE's is given by nonlinear Robin boundary conditions. This article provides two major new contributions regarding these infiltration conditions. First, an existence result for the continuous coupled problem is established with the help of a regularization technique. Second, we analyze and validate a solver-friendly discretization of the coupled problem based on an implicit-explicit time discretization and on finite elements in space. The discretized PDE leads to convex spatial minimization problems which can be solved efficiently by monotone multigrid. Numerical experiments are provided using the DUNE numerics framework.

math.NA↗