Inverse heat conduction problems (IHCPs) have been extensively studied over the last 50 years. They have numerous applications in many branches of science and technology. The problem consists in determining the temperature and heat flux at inaccessible parts of the boundary of a 2-or 3-dimensional body from corresponding data -called 'Cauchy data' -on accessible parts of the boundary. It is well known that IHCPs are severely illposed which means that small perturbations in the data may cause extremely large errors in the solution. In this contribution we first present the problem and show examples of calculations for 2-dimensional IHCP's where the direct problems are solved with the Finite Element package DEAL. As solution procedure we use Tikhonov's regularization in combination with the conjugate gradient method.Key words. inverse heat conduction problem, regularization, noncharacteristic Cauchy problem for parabolic equations, finite element method.
Abstract. Elastoplastic contact problems with hardening are ubiquitous in industrial metal forming processes as well as many other areas. From a mathematical perspective, they are characterized by the difficulties of variational inequalities for both the plastic behavior as well as the contact problem. Computationally, they also often lead to very large problems. In this paper, we present and evaluate a set of methods that allows us to efficiently solve such problems. In particular, we use adaptive finite element meshes with linear and quadratic elements, a Newton linearization of the plasticity, active set methods for the contact problem, and multigrid-preconditioned linear solvers. Through a sequence of numerical experiments, we show the performance of these methods. This includes highly accurate solutions of a benchmark problem and scaling our methods to 1,024 cores and more than a billion unknowns.
In this article a semi-smooth Newton method for frictional two-body contact problems and a solution algorithm for the resulting sequence of linear systems are presented. It is based on a mixed variational formulation of the problem and a discretization by finite elements of higher-order. General friction laws depending on the normal stresses and elasto-plastic material behavior with linear isotropic hardening are considered. Numerical results show the efficiency of the presented algorithm.
In this paper our studies on techniques for a posteriori error control and adaptive mesh design for finite element models in perfect plasticity are continued. The focus is on the numerical analysis of a low-order, dual-mixed discretisation. A posteriori error estimates are provided. Numerical tests confirm the theoretical results.
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