peter gruber - Academia.edu (original) (raw)

Papers by peter gruber

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2010

In this paper some analytical and numerical aspects of time-dependent models with internal variab... more In this paper some analytical and numerical aspects of time-dependent models with internal variables are discussed. The focus lies on elasto/visco-plastic models of monotone type arising in the theory of inelastic behavior of materials. This class of problems includes the classical models of elasto-plasticity with hardening and viscous models of the Norton-Hoff type. We discuss the existence theory for different models of monotone type, give an overview on spatial regularity results for solutions to such models and illustrate a numerical solution algorithm at an example. Finally, the relation to the energetic formulation for rate-independent processes is explained and temporal regularity results based on different convexity assumptions are presented.

Dimensions, 1995

This work is concerned with the numerical solution of variational problems in elastoplasticity. S... more This work is concerned with the numerical solution of variational problems in elastoplasticity. Since the whole class of all possible elastoplastic problems is too large for a common treatment, we restrict ourselves to the investigation of problems which are quasi-static, and where strain and displacement are related linearly. Further, the homogeneous and isotropic material should obey the Prandtl-Reuß flow and a linear hardening principle. After an implicit Euler discretization in time, it is possible to derive a one-time step minimization problem, which can be solved by a Newton-like method. Each iteration step represents a linear boundary value problem, which has to be discretized in space and approximately solved on the computer. Due to regularity reasons, the application of a Finite Element Method (FEM) of low and high order (hp-FEM) looks promising. Roughly speaking, low order FEM is used in regions of the domain where the solution is expected to have low regularity (plastic zones), while the use of high order FEM speeds up the convergence of the approximate solution in regions where the solution has high regularity (elastic zones). We discuss a several different hp-adaptive Strategies, particularly a technique is applied, where the regularity of the solution is estimated by analyzing the expansion of the Finite Element solution with respect to an L 2-orthogonal polynomial basis. Also the application of a Boundary Concentrated FEM to elastoplastic problems is discussed and numerically tested.

Newton-Like Solver for Elastoplastic Problems with Hardening and its Local Super-Linear Convergence

Numerical Mathematics and Advanced Applications

We discuss a new solution algorithm for solving elastoplastic problems with hardening. The one ti... more We discuss a new solution algorithm for solving elastoplastic problems with hardening. The one time-step elastoplastic problem can be formulated as a convex minimization problem with a continuous but non-smooth functional dependening on unknown displacement smoothly and on the plastic strain non-smoothly. It is shown that the functional structure allows the application of the Moreau-Yosida Theorem known in convex analysis. It guarantees that the substitution of the non-smooth plastic-strain as a function of the linear strain which depends on the displacement only yields an already smooth functional in the displacement only. Moreover, the second derivative of such functional exists in all continuum points apart from interfaces where elastic and plastic zones intersect. This allows the efficient implementation of the Newton-Ralphson method. For easy implementation most essential Matlab c functions are provided. Numerical experiments in two dimensions state quadratic convergence of a Newton-Ralpshon method as long as the elastoplastic interface is detected sufficiently precisely.

Solution of One-Time-Step Problems in Elastoplasticity by a Slant Newton Method

SIAM Journal on Scientific Computing, 2009

Fast Solvers and A Posteriori Error Estimates in Elastoplasticity

Texts & Monographs in Symbolic Computation, 2011

Mathematics and Computers in Simulation, 2007

We discuss a technique for solving elastoplastic problems with hardening. The one time-step elast... more We discuss a technique for solving elastoplastic problems with hardening. The one time-step elastoplastic problem can be formulated as a convex minimization problem with a continuous but non-smooth objective. We actually show that its objective structure satisfies conditions of the Moreau-Yosida Theorem known from convex analysis. Therefore, the substitution of the nonsmooth plastic-strain p as a function of the total strain ε(u) yields an already smooth functional in the displacement u only. The second derivative of such functional exists in all continuum points apart from interfaces where elastic and plastic zones intersect. The numerical experiment states super-linear convergence of a Newton method or even quadratic convergence as long as the interface is detected sufficiently.

Interaction of flexible multibody systems with fluids analyzed by means of smoothed particle hydrodynamics

Multibody System Dynamics, 2013

ABSTRACT The present work deals with a computational approach to fluid-structure interaction (FSI... more ABSTRACT The present work deals with a computational approach to fluid-structure interaction (FSI) problems by coupling of flexible multibody system dynamics and fluid dynamics. Since the methods for the numerical modeling are well known, both for the structural and the fluid part, the focus of this work lies on the coupling formalism. Moreover, the applicability of the presented approach to arbitrary geometries and high structural stiffness is studied, as well as an easy model setup. No restriction should be made on the topology of the structure or the complexity of motion. For the fluid part a meshless method, known as smoothed particle hydrodynamics (SPH) is applied, which fulfills the above requirements. While an explicit time integration scheme in SPH provides a fast simulation of the fluid dynamics, advanced methods from flexible multibody dynamics provide a variety of benefits for the simulation of the solid part. Amongst these are specialized structural finite elements for both small and large deformation bodies, joints, stable implicit time-integration schemes, and model reduction techniques. A rule for the interaction between fluids and structures is derived from imposing a distributed potential over boundary segments of the structures, which the fluid particles respond to. The work is concluded by illustrative examples, demonstrating the successful coupling of flexible multibody systems with fluids.