474 research outputs found
New Relaxation Modulus Based Iterative Method for Large and Sparse Implicit Complementarity Problem
This article presents a class of new relaxation modulus-based iterative
methods to process the large and sparse implicit complementarity problem (ICP).
Using two positive diagonal matrices, we formulate a fixed-point equation and
prove that it is equivalent to ICP. Also, we provide sufficient convergence
conditions for the proposed methods when the system matrix is a -matrix or
an -matrix.
Keyword: Implicit complementarity problem, -matrix, -matrix, matrix
splitting, convergenceComment: arXiv admin note: substantial text overlap with arXiv:2303.1251
Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming
Cover title.Includes bibliographical references.Partially supported by the U.S. Army Research Office (Center for Intelligent Control Systems) DAAL03-86-K-0171 Partially supported by the National Science Foundation. NSF-ECS-8519058Paul Tseng
Forward-backward truncated Newton methods for convex composite optimization
This paper proposes two proximal Newton-CG methods for convex nonsmooth
optimization problems in composite form. The algorithms are based on a a
reformulation of the original nonsmooth problem as the unconstrained
minimization of a continuously differentiable function, namely the
forward-backward envelope (FBE). The first algorithm is based on a standard
line search strategy, whereas the second one combines the global efficiency
estimates of the corresponding first-order methods, while achieving fast
asymptotic convergence rates. Furthermore, they are computationally attractive
since each Newton iteration requires the approximate solution of a linear
system of usually small dimension
Some recent advances in projection-type methods for variational inequalities
AbstractProjection-type methods are a class of simple methods for solving variational inequalities, especially for complementarity problems. In this paper we review and summarize recent developments in this class of methods, and focus mainly on some new trends in projection-type methods
A modified combined active-set Newton method for solving phase-field fracture into the monolithic limit
In this work, we examine a numerical phase-field fracture framework in which
the crack irreversibility constraint is treated with a primal-dual active set
method and a linearization is used in the degradation function to enhance the
numerical stability. The first goal is to carefully derive from a
complementarity system our primal-dual active set formulation, which has been
used in the literature in numerous studies, but for phase-field fracture
without its detailed mathematical derivation yet. Based on the latter, we
formulate a modified combined active-set Newton approach that significantly
reduces the computational cost in comparison to comparable prior algorithms for
quasi-monolithic settings. For many practical problems, Newton converges fast,
but active set needs many iterations, for which three different efficiency
improvements are suggested in this paper. Afterwards, we design an iteration on
the linearization in order to iterate the problem to the monolithic limit. Our
new algorithms are implemented in the programming framework pfm-cracks [T.
Heister, T. Wick; pfm-cracks: A parallel-adaptive framework for phase-field
fracture propagation, Software Impacts, Vol. 6 (2020), 100045]. In the
numerical examples, we conduct performance studies and investigate efficiency
enhancements. The main emphasis is on the cost complexity by keeping the
accuracy of numerical solutions and goal functionals. Our algorithmic
suggestions are substantiated with the help of several benchmarks in two and
three spatial dimensions. Therein, predictor-corrector adaptivity and parallel
performance studies are explored as well.Comment: 49 pages, 45 figures, 9 table
Phase-field fracture modeling, numerical solution, and simulations for compressible and incompressible solids
In this thesis, we develop phase-field fracture models for simulating fractures in compressible and incompressible solids. Classical (primal) phase-field fracture models fail due to locking effects. Hence, we formulate the elasticity part of the phase-field fracture problem in mixed form, avoiding locking. For the elasticity part in mixed form, we prove inf-sup stability, which allows a stable discretization with Taylor-Hood elements. We solve the resulting (3x3) phase-field fracture problem - a coupled variational inequality system - with a primal-dual active set method. In addition, we develop a physics-based Schur-type preconditioner for the linear solver to reduce the computational workload. We confirm the robustness of the new solver for five benchmark tests. Finally, we compare numerical simulations to experimental data analyzing fractures in punctured strips of ethylene propylene diene monomer rubber (EPDM) stretched until total failure to check the applicability on a real-world problem in nearly incompressible solids. Similar behavior of measurement data and the numerically computed quantities of interest validate the newly developed quasi-static phase-field fracture model
in mixed form.DFG/SPP 1748/392587580/E
Modeling, Discretization, Optimization, and Simulation of Phase-Field Fracture Problems
This course is devoted to phase-field fracture methods. Four different sessions are centered around modeling, discretizations, solvers, adaptivity, optimization, simulations and current developments. The key focus is on research work and teaching materials concerned with the accurate, efficient and robust numerical modeling. These include relationships of model, discretization, and material parameters and their influence on discretizations and the nonlinear (Newton-type methods) and linear numerical solution. One application of such high-fidelity forward models is in optimal control, where a cost functional is minimized by controlling Neumann boundary conditions. Therein, as a side-project (which is itself novel), space-time phase-field fracture models have been developed and rigorously mathematically proved. Emphasis in the entire course is on a fruitful mixture of theory, algorithmic concepts and exercises. Besides these lecture notes, further materials are available, such as for instance the open-source libraries pfm-cracks and DOpElib.
The prerequisites are lectures in continuum mechanics, introduction to numerical methods, finite elements, and numerical methods for ODEs and PDEs. In addition, functional analysis (FA) and theory of PDEs is helpful, but for most parts not necessarily mandatory.
Discussions with many colleagues in our research work and funding from the German Research Foundation within the Priority Program 1962 (DFG SPP 1962) within the subproject Optimizing Fracture Propagation using a Phase-Field Approach with the project number 314067056 (D. Khimin, T. Wick), and support of the French-German University (V. Kosin) through the French-German Doctoral college ``Sophisticated Numerical and Testing Approaches" (CDFA-DFDK 19-04) is gratefully acknowledged
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