An rp-adaptive finite element method for the deformation theory of plasticity

In this paper, we present an rp-discretization strategy for physically non-linear problems based on a high order finite element formulation. In order to achieve convergence, the p-version leaves the mesh unchanged and increases the polynomial degree of the shape functions locally or globally, whereas the r-method moves nodes and edges of an existing FE-mesh. The basic idea of our rp-version approach is to adjust the finite element mesh to the shape of the elastic–plastic interface in order to take into account the loss of regularity which arises along the curve of the plastic front. Numerical examples will demonstrate that this approach leads to an exponential rate of convergence and highly accurate results.

[1]  Ernst Rank,et al.  A p‐version finite element approach for two‐ and three‐dimensional problems of the J2 flow theory with non‐linear isotropic hardening , 2002 .

[2]  Z. Yosibash,et al.  THE p-VERSION OF THE FINITE ELEMENT METHOD IN INCREMENTAL ELASTO-PLASTIC ANALYSIS , 1993 .

[3]  Ernst Rank,et al.  The p‐version of the finite element method for three‐dimensional curved thin walled structures , 2001 .

[4]  C. Schwab.  P- and hp- finite element methods : theory and applications in solid and fluid mechanics , 1998 .

[5]  W. J. Gordon,et al.  Construction of curvilinear co-ordinate systems and applications to mesh generation , 1973 .

[6]  H. Broeker,et al.  Integration von geometrischer Modellierung und Berechnung nach der p-Version der FEM , 1999 .

[7]  E. Stein,et al.  Error indicators and mesh refinements for finite-element-computations of elastoplastic deformations , 1998 .

[8]  H. Hencky,et al.  Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannungen , 1924 .

[9]  Ernst Rank,et al.  A Numerical Investigation of High-Order Finite Elements for Problems of Elastoplasticity , 2002, J. Sci. Comput..

[10]  Ernst Rank,et al.  The p-version of the finite element method compared to an adaptive h-version for the deformation theory of plasticity , 2001 .

[11]  Ivo Babuška,et al.  Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle , 1995 .

[12]  Spannungsverteilung in plastischen Körpern , 1961 .

[13]  Ernst Rank,et al.  The p‐version of the finite element method for domains with corners and for infinite domains , 1990 .

[14]  István Szabó.  Höhere Technische Mechanik , 1956 .

[15]  Christos Xenophontos,et al.  Application of the p‐version of the finite element method to elastoplasticity with localization of deformation , 1999 .

[16]  Ernst Rank,et al.  On the accuracy of p-version elements for the Reissner-Mindlin plate problem , 1998 .

[17]  R. Actis,et al.  Solution of elastic-plastic stress analysis problems by the P-version of the finite element method , 1995 .

[18]  Barna A. Szabó,et al.  Quasi-regional mapping for the p-version of the finite element method , 1997 .

[19]  J. Chaboche,et al.  Mechanics of Solid Materials , 1990 .

[20]  I. Babuska,et al.  Finite Element Analysis , 2021 .

[21]  J. C. Rice,et al.  On numerically accurate finite element solutions in the fully plastic range , 1990 .

[22]  Franz-Joseph Barthold,et al.  Adaptive finite elements in elastoplasticity with mechanical error indicators and Neumann-type estimators , 1998 .

[23]  Ivo Babuška,et al.  The optimal symmetrical points for polynomial interpolation of real functions in the tetrahedron , 1995 .

[24]  W. J. Gordon,et al.  Transfinite element methods: Blending-function interpolation over arbitrary curved element domains , 1973 .

[25]  E. Ramm,et al.  Error-controlled Adaptive Finite Elements in Solid Mechanics , 2001 .