Rainer Niekamp

MULTI-SCALE MODELLING OF HETEROGENEOUS STRUCTURES WITH INELASTIC CONSTITUTIVE BEHAVIOR

Purpose – To provide a computational strategy for highly accurate analyses of non‐linear inelastic behaviour for heterogeneous structures in civil and mechanical engineering applications Design/methodology/approach – Adapts recent developments on mathematical formulations of multi‐scale problems to the recently developed component technology based on C++ generic templates programming. Findings – Provides the understanding how theoretical hypotheses, concerning essentially the multi‐scale interface conditions, affect the computational precision of the strategy. Practical implications – The present approach allows a very precise modelling of multi‐scale aspects in structural mechanics problems and can play an essential tool in searching for an optimal structural design. Originality/value – Provides all the ingredients for constructing an efficient multi‐scale computational framework, from the theoretical formulation to the implementation for parallel computing. It is addressed to researchers and engineers analysing composite structures under extreme loading.

Error indicators for mixed finite elements in 2-dimensional linear elasticity

Abstract We establish a posteriori error indicators for mixed finite elements in plane elasticity. The error estimators refer to residuals of the strong equations and to jumps of the displacements on interelement boundaries. For the BDM elements of lowest order, the error indicators are computed with displacement fields which are obtained by a postprocessing procedure. Numerical examples show that adaptive mesh refinements based on these estimators lead to very efficient algorithms.

An object-oriented approach for parallel two- and three-dimensional adaptive finite element computations

For the mathematically sound, cost effective, flexible and automatic computation of structural mechanical problems with error tolerances, adaptive finite element meshes (h-adaptivity) and elements with different Ansatz order (p-adaptivity) and dimension (d-adaptivity) are desirable. Furthermore, because of the numerical effort, the use of parallel computers is adequate. Object-oriented data structures and algorithms are presented which support these adaptive formulations. In this paper, we describe a refinement algorithm which adapts hexahedral meshes in a node regular way, i.e. without hanging nodes. Moreover, classes implementing the mathematical operators and structures arising in the finite element formulation are introduced within the object oriented concept. They offer the means to implement FE formulations in a way, very similar to the mathematical notation. For these purposes, an object-oriented language is strongly required in order to get a general and simple program structure, even for highly complex tasks with h-, p- and d-adaptivity and distributed data.

Partitioned treatment of uncertainty in coupled domain problems: A separated representation approach

Abstract This work is concerned with the propagation of uncertainty across coupled domain problems with high-dimensional random inputs. A stochastic model reduction approach based on low-rank separated representations is proposed for the partitioned treatment of the uncertainty space. The construction of the coupled domain solution is achieved though a sequence of approximations with respect to the dimensionality of the random inputs associated with each individual sub-domain and not the combined dimensionality, hence drastically reducing the overall computational cost. The coupling between the sub-domain solutions is done via the classical finite element tearing and interconnecting (FETI) method, thus providing a well suited framework for parallel computing. Two high-dimensional stochastic problems, a 2D elliptic PDE with random diffusion coefficient and a stochastic linear elasticity problem, have been considered to study the performance and accuracy of the proposed stochastic coupling approach.

Multi‐scale modelling of heterogeneous structures with inelastic constitutive behavior: Part II – software coupling implementation aspects

Purpose – The purpose of this paper is to consider the computational tools for solving a strongly coupled multi‐scale problem in the context of inelastic structural mechanics. Design/methodology/approach – In trying to maintain the highest level of generality, the finite element method is employed for representing the microstructure at this fine scale and computing the solution. The main focus of this work is the implementation procedure which crucially relies on a novel software product developed by the first author in terms of component template library (CTL). Findings – The paper confirms that one can produce very powerful computational tools by software coupling technology described herein, which allows the class of complex problems one can successfully tackle nowadays to be extended significantly. Originality/value – This paper elaborates upon a new multi‐scale solution strategy suitable for highly non‐linear inelastic problems.

On Three-Dimensional Microcrack Density Distribution

This paper considers tensorial representations of several microerack distribution functions due to tensile and compressive principal stresses in brittle materials in the framework of continuum mechanics. The common framework for deriving the damage tensors of different order from any density function is suggested. Second and fourth order damage tensors are derived for Dirac-. truncated Gauss-, and trigonometrical (cos 2 -) microcrack distributions using harmonic Fourier-like series. Each distribution is investigated under different combinations of tensile and compressive principal stresses for three-dimensional load cases. It is emphasized that only the trigonometrical distribution yields a spherical crack density surface for the fourth order tensor approximation under three equal principal stresses.

Continuous and discontinuous Galerkin methods with finite elements in space and time for parallel computing of viscoelastic deformation

Abstract New non-symmetric variational and discretized formulations (with space-time finite elements) are proposed for viscoelastic problems based on the continuous Galerkin method (CGM) and discontinuous Galerkin method (DGM). Viscoelastic behaviour is described by the three-parameter Malvern model, which is represented by means of internal variables. It allows to use only differential equations for the constitutive equations instead of integrodifferential ones. The variational formulation reduces to two types of equations for total displacements and internal displacements (internal variables), namely to the equilibrium equation and the evolution equation for the internal displacements, which are fulfilled in the weak form. Using continuous trial functions, a continuous space-time finite element formulation is obtained with simultaneous discretization in space and time. Subdividing the total observation time interval into time slabs and introducing discontinuous trial functions, being continuous within time slabs and allowing jumps across interfaces, a more general discontinuous finite element formulation is obtained. The difference between these two formulations for one time slab consists in the satisfaction of initial conditions which are fulfilled exactly for the continuous formulation and in a weak form for the discontinuous case. The proposed approach has some very attractive advantages with respect to semidiscretization methods, regarding the possibility of adaptive space-time refinements and parallel processing on MIMD-parallel computers. The considered numerical examples show the effectiveness of simultaneous space-time finite element calculations and a high convergence rate for adaptive refinement. Numerical efficiency is an advantage of DGM in comparison with CGM for discontinuously changing (e.g. piecewise constant) boundary conditions in time.

Parallel adaptive finite element computations with hierarchical preconditioning

A parallel implementation of an adaptive finite element program is treated which is characterized by an underlying parallel dynamic data structure based on linked lists and tree structures. In conjunction with a conjugate gradient solver an efficient methodology for treating adaptive finite element systems is shown. This is achieved by preconditioning using hierarchical bases with and without a coarse grid solver and by new methods of quasi-optimal load balancing. The different levels of nested meshes needed for preconditioning are governed either by global or by adaptive refinements. A termination algorithm based on the vector method is implemented for the non deterministic adaptive mesh refinement procedure. The problems concerning load balancing due to adaptive refinement are solved by a dynamic load balancing for the nodes.

Coupling of adaptively refined dual mixed finite elements and boundary elements in linear elasticity

We investigate a coupling of mixed finite elements and Galerkin boundary elements which is stable and leads to symmetric matrices. In the FEM domain, a posteriori error estimates are employed to refine the mesh adaptively. Numerical results are given for plane strain problems.

Code-coupling strategy for efficient development of computer software in multiscale and multiphysics nonlinear evolution problems in computational mechanics

Abstract In this work we seek to provide an efficient approach to development of software computational platform for the currently very active research domain of multiphysics and multiscale analysis in fully nonlinear setting. The typical problem to be solved is nonlinear evolution problem, with different scales in space and time. We show here that a successful solution to such a problem requires gathering the sound theoretical formulation, the most appropriate discrete approximation and the efficient numerical implementation. We show in particular that the most efficient numerical implementation is obtained by reusing the existing codes, in order to accelerate the code development and validation. The key element that makes such an approach possible is the Component Template Library (CTL), presented in this work. We show that the CTL allows to seamlessly merge the existing software products into a single code at compilation time, regardless of their ‘heterogeneities’ in terms of programming language or redundancy in use of local variables. A couple of illustrative problems of fluid–structure interaction and multiscale nonlinear analysis are presented in order to confirm the advantage of the proposed approach.