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- geodesic finite elements (4)
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We present an algorithm that constructs parametrizations of boundary
and interface surfaces automatically. Starting with high-resolution triangulated
surfaces describing the computational domains, we iteratively
simplify the surfaces yielding a coarse approximation of the boundaries
with the same topological type. While simplifying we construct a function
that is defined on the coarse surface and whose image is the original
surface. This function allows access to the correct shape and surface normals
of the original surface as well as to any kind of data defined on it.
Such information can be used by geometric multigrid solvers doing adaptive
mesh refinement. Our algorithm runs stable on all types of input
surfaces, including those that describe domains consisting of several materials.
We have used our method with success in different fields and we
discuss examples from structural mechanics and biomechanics.
We present globally convergent multigrid methods for the nonsymmetric
obstacle problems as arising from the discretization of Black–Scholes models of
American options with local volatilities and discrete data. No tuning or regularization
parameters occur. Our approach relies on symmetrization by transformation
and data recovery by superconvergence.
We present a new solver for large-scale two-body contact problems in nonlinear elasticity. It is based on an SQP-trust-region approach.
This guarantees global convergence to a first-order critical point of
the energy functional. The linearized contact conditions are
discretized using mortar elements. A
special basis transformation known from linear contact problems
allows to use a monotone multigrid solver for the inner quadratic programs.
They can thus be solved with multigrid complexity. Our algorithm
does not contain any regularization or penalization parameters,
and can be used for all hyperelastic material models.
The purpose of the paper is to apply monotone multigrid methods
to static and dynamic biomechanical contact problems.
In space, a finite element method involving a mortar
discretization of the contact conditions is used.
In time, a new contact--stabilized Newmark scheme is presented.
Numerical experiments for a two body Hertzian contact problem
and a biomechanical knee problem are reported.
We present a new inexact nonsmooth Newton method for the solution
of convex minimization problems with piecewise smooth, pointwise
nonlinearities. The algorithm consists of a nonlinear smoothing
step on the fine level and a linear coarse correction.
Suitable postprocessing guarantees global convergence
even in the case of a single multigrid step for each linear subproblem.
Numerical examples show that the overall efficiency
is comparable to multigrid for similar linear problems.
The PSurface Library
(2010)
We describe psurface, a C++ library that allows to store and access piecewise linear mappings between simplicial surfaces in $\R^2$ and $\R^3$. These mappings are stored in a graph data structure and can be constructed explicitly, by projection, or by surface simplification. Piecewise linear maps can be used, e.g., to construct boundary
approximations for finite element grids, and grid intersections for domain decomposition methods. In computer graphics the mappings allow to build level-of-detail representations as well as texture- and bump maps. We document the data structures and algorithms used and show how \psurface is used in the numerical analysis framework Dune
and the visualization software Amira.
We prove global convergence of an inexact polyhedral Gau\ss--Seidel method for the minimization of strictly convex functionals that are continuously differentiable on each polyhedron of a polyhedral decomposition of
their domains of definition. While being known to be very slow by themselves, such methods are a cornerstone for fast, globally convergent multigrid methods. Our result generalizes the proof of Kornhuber and Krause [2006] for differentiable functionals on the Gibbs simplex. Example applications are given that require the generality of our approach.
We investigate geodesic finite elements for functions with values in a space of zero curvature, like a torus or the M\"obius strip. Unlike in the general case, a closed-form expression for geodesic finite element functions is then available. This simplifies computations, and allows us to prove optimal estimates for the interpolation error in 1d and 2d. We also show the somewhat surprising result that the discretization by Kirchhoff transformation of the Richards equation proposed by Berninger et al. is a discretization by geodesic finite elements in the manifold $\mathbb{R}$ with a special metric.
We consider the mechanical coupling of a geometrically exact Cosserat rod to a linear elastic continuum. The coupling conditions are formulated in the nonlinear rod configuration space. We describe a Dirichlet--Neumann algorithm for the coupled system, and use it to simulate the static stresses in a human knee joint, where the Cosserat rods are models for the ligaments.
We introduce geodesic finite elements as a conforming way to discretize partial differential equations for functions $v : \Omega \to M$, where $\Omega$ is an open subset of $\R^d$ and $M$ is a Riemannian manifold. These geodesic finite elements naturally generalize standard first-order
finite elements for Euclidean spaces. They also generalize the geodesic finite elements proposed for $d=1$ by the author. Our formulation is equivariant under isometries of $M$, and hence preserves objectivity of continuous problem formulations. We concentrate on partial differential equations that can be formulated as minimization problems. Discretization leads to algebraic minimization problems on product manifolds $M^n$. These can be solved efficiently
using a Riemannian trust-region method. We propose a monotone multigrid method to solve the constrained inner problems with linear multigrid speed. As an example we numerically compute harmonic maps from a domain in $\R^3$
to $S^2$.
We formulate the static mechanical coupling of a geometrically exact Cosserat rod
to a nonlinearly elastic continuum. In this setting, appropriate coupling conditions have
to connect a one-dimensional model with director variables to a three-dimensional
model without directors.
Two alternative coupling conditions are proposed,
which correspond to two different configuration trace spaces.
For both we show existence of solutions of the coupled problems, using the direct
method of the calculus of variations. From the first-order optimality conditions
we also derive the corresponding conditions for the dual variables. These are
then interpreted in mechanical terms.
We present a time-dependent finite element model of the human knee joint of full 3D geometric complexity together with advanced numerical algorithms needed for its simulation. The model comprises bones, cartilage and the major ligaments, while patella and menisci are still missing. Bones are modeled by linear elastic materials, cartilage by linear viscoelastic materials, and ligaments by one-dimensional nonlinear Cosserat rods. In order to capture the dynamical contact problems correctly, we solve the full PDEs of elasticity with strict contact inequalities. The spatio--temporal discretization follows a time layers approach (first time, then space discretization). For the time discretization of the elastic and viscoelastic parts we use a new contact-stabilized Newmark method, while for the Cosserat rods we choose an energy--momentum method. For the space discretization, we use linear finite elements for the elastic and viscoelastic parts and novel geodesic finite elements for the Cosserat rods. The coupled system is solved by a Dirichlet--Neumann method. The large algebraic systems of the bone--cartilage contact problems are solved efficiently by the truncated non-smooth Newton multigrid method.
We present a discretization for dynamic large deformation contact problems without friction. Our model is based on Hamilton’s principle, which avoids the explicit appearance of the contact forces. The resulting differential inclusion is discretized in time using a modified midpoint rule. This modification, which concerns the evaluation of the generalized gradient, allows to achieve energy dissipativity. For the space discretization we use a dual-basis mortar method. The resulting spatial algebraic
problems are nonconvex minimization problems with nonconvex inequality constraints. These can be solved efficiently using a trust-region SQP framework with a monotone multigrid inner solver.
A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part I: Abstract Framework
(2007)
We give a mathematically rigorous definition of a grid for algorithms solving
partial differential equations. Unlike previous approaches, our grids have a
hierarchical structure. This makes them suitable for geometric multigrid
algorithms and hierarchical local grid refinement. The description is also
general enough to include geometrically nonconforming grids. The definitions
in this article serve as the basis for an implementation of an abstract grid
interface as C++ classes in the DUNE.
In a companion paper [Matheon-Preprint 403] we introduced an abstract definition of a parallel and adaptive hierarchical grid for scientific computing. Based on this
definition we derive an efficient interface specification as a set of C++ classes.
This interface separates the applications from the grid data structures.
Thus, user implementations become independent of the underlying grid
implementation. Modern C++ template techniques are used to provide an
interface implementation without big performance losses.
The implementation is realized as part of the
software environment DUNE.
Numerical tests demonstrate the flexibility and the efficiency of our approach.
We present an extension module for the Dune system. This module, called dune-subgrid, allows to mark elements of another Dune hierarchical grid. The set of marked elements can then be accessed as a Dune grid in its own right. dune-subgrid is free software and is available for download. We describe the functionality and use of dune-subgrid, comment on its implementation, and give two example applications.
First, we show how dune-subgrid can be used for micro-FE simulations of trabecular bone. Then we present an algorithm that allows to use exact residuals for the adaptive solution of the spatial problems of time-discretized evolution equations.
We introduce geodesic finite elements as a new way to discretize
the nonlinear configuration space of a geometrically exact Cosserat rod.
These geodesic finite elements naturally generalize standard one-dimensional
finite elements to spaces of functions with values in a Riemannian manifold.
For the special orthogonal group, our approach reproduces the
interpolation formulas of [Crisfield/Jelenic:1999].
Geodesic finite elements are
conforming and lead to objective and path-independent problem formulations.
We introduce geodesic finite elements for general Riemannian manifolds,
discuss the relationship between geodesic finite elements and
coefficient vectors, and estimate the interpolation error.
Then we use them to find static equilibria of hyperelastic Cosserat rods.
Using the Riemannian trust-region algorithm of [Absil/Mahony/Sepulchre:2008]
we show numerically that the discretization error depends optimally on
the mesh size.
We formulate the static mechanical coupling of a geometrically exact Cosserat rod to an elastic continuum. The coupling conditions accommodate for the difference in dimension between the two models. Also, the Cosserat rod model incorporates director variables, which are not present in the elastic continuum model. Two alternative coupling conditions are proposed, which correspond to two different configuration trace spaces. For both we show existence of solutions of the coupled problems. We also derive the corresponding conditions for the dual variables and interpret them in mechanical terms.
We present a globally convergent method for the solution of frictionless large deformation contact problems involving hyperelastic materials.
For the discretisation we apply the dual 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.
This method combines several techniques from non-linear optimisation to achieve global convergence towards first-order optimal points. To speed up the method inexact linearisations of the non-penetration constraint are used
whenever the current iterate is far away from a critical point. A monotone multigrid method is applied for the fast solution of the constrained Newton
problems.