Adaptive Finite Elements and Mathematical Optimization Methods

This chapter focuses on the topics of mathematical research in the priority program and on the application of the results to engineering problems. First, a posteriori estimates and adaptive methods based on them are presented for contact problems. In detail, frictional contact problems using a linear elastic material law, which are discretized by a dual-dual method, are considered. Using a different approach, adaptive methods are derived for elasto-plasticity with contact and friction. Then results for contact problems involving inertial effects are introduced. Second, parameter identification and inverse problems are considered. After an introduction into the general problem of setting and solution techniques, the balancing of a rotating system is discussed as a prototypical industrial application.

[1]  J. Douglas,et al.  PEERS: A new mixed finite element for plane elasticity , 1984 .

[2]  Andreas Schröder,et al.  Error estimates in elastoplasticity using a mixed method , 2011 .

[3]  Andreas Schröder,et al.  Space adaptive finite element methods for dynamic obstacle problems. , 2008 .

[4]  Weimin Han,et al.  A posteriori error estimation and adaptive solution of elliptic variational inequalities of the second kind , 2005 .

[5]  Ronny Ramlau,et al.  A steepest descent algorithm for the global minimization of the Tikhonov functional , 2002 .

[6]  Alexandre Ern,et al.  A Posteriori Control of Modeling Errors and Discretization Errors , 2003, Multiscale Model. Simul..

[7]  Ronny Ramlau,et al.  TIGRA—an iterative algorithm for regularizing nonlinear ill-posed problems , 2003 .

[8]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[9]  Michael Andres.  Dual-dual formulations for frictional contact problems in mechanics , 2010 .

[10]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[11]  Carsten Carstensen,et al.  Reliable and efficient equilibrated a posteriori finite element error control in elastoplasticity and elastoviscoplasticity with hardening , 2006 .

[12]  Andreas Schröder,et al.  Goal-oriented error control in adaptive mixed FEM for Signorini's Problem , 2011 .

[13]  R. Glowinski,et al.  Numerical Analysis of Variational Inequalities , 1981 .

[14]  Carsten Carstensen,et al.  A posteriori error estimates for mixed FEM in elasticity , 1998, Numerische Mathematik.

[15]  C. Carstensen,et al.  A quasi‐static boundary value problem in multi‐surface elastoplasticity: part 2—numerical solution , 2005 .

[16]  Carsten Carstensen,et al.  A quasi‐static boundary value problem in multi‐surface elastoplasticity: Part 1—Analysis , 2004 .

[17]  Ronny Ramlau,et al.  Modeling the influence of unbalances for ultra‐precision cutting processes , 2011 .

[18]  R. Ramlau.  MOROZOV'S DISCREPANCY PRINCIPLE FOR TIKHONOV-REGULARIZATION OF NONLINEAR OPERATORS , 2002 .

[19]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[20]  Otmar Scherzer,et al.  The use of Morozov's discrepancy principle for Tikhonov regularization for solving nonlinear ill-posed problems , 1993, Computing.

[21]  Andreas Schröder.  A Posteriori Error Estimation in Mixed Finite Element Methods for Signorini’s Problem , 2010 .

[22]  T. Laursen.  Computational Contact and Impact Mechanics , 2003 .

[23]  Rolf Rannacher,et al.  A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity , 1999 .

[24]  Andreas Schröder.  Mixed Finite Element Methods of Higher-Order for Model Contact Problems , 2011, SIAM J. Numer. Anal..

[25]  S. Yau.  Mathematics and its applications , 2002 .

[26]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[27]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis: Oden/A Posteriori , 2000 .

[28]  I. S. Sokolnikoff.  Mathematical theory of elasticity , 1946 .

[29]  Andreas Schröder,et al.  Space adaptive finite element methods for dynamic Signorini problems , 2009 .

[30]  Weimin Han.  Finite element analysis of a holonomic elastic-plastic problem , 1991 .

[31]  Andreas Schröder.  Fehlerkontrollierte adaptive h- und hp-Finite-Elemente-Methoden für Kontaktprobleme mit Anwendungen in der Fertigungstechnik , 2006 .

[32]  Andreas Rademacher,et al.  Finite elements in space and time for dynamic contact problems , 2008 .

[33]  Almerico Murli,et al.  Numerical Mathematics and Advanced Applications , 2003 .

[34]  Andreas Rademacher,et al.  Adaptive finite element methods for nonlinear hyperbolic problems of second order , 2010 .

[35]  Carsten Carstensen,et al.  Numerical analysis of the primal problem of elastoplasticity with hardening , 1999, Numerische Mathematik.

[36]  Shaker A. Meguid,et al.  On the elastodynamic solution of frictional contact problems using variational inequalities , 2001 .

[37]  Harold A. Buetow,et al.  Adaptive finite element methods for differential equations , 2003, Lectures in mathematics.

[38]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[39]  Kazufumi Ito,et al.  Lagrange multiplier approach to variational problems and applications , 2008, Advances in design and control.

[40]  J. Haslinger,et al.  Solution of Variational Inequalities in Mechanics , 1988 .