Hermann G. Matthies

Uncertainties in probabilistic numerical analysis of structures and solids-Stochastic finite elements

The main sources of uncertainties involved in the analysis of structures and solids are shown and the tools available to deal with them. While trying to cover the complete modeling process, ranging from the problem formulation via the mathematical model all the way to the numerical approximation, we have tried to expose areas in need of further research. The techniques and methods involved in stochastic modeling are explained in somewhat more detail, as they are newer and less known than those used for the deterministic modeling.

Partitioned Strong Coupling Algorithms for Fluid-Structure-Interaction

Abstract Numerical simulation of fluid–structure interaction is often attempted in the context of partitioned methods, where already existing solvers for fluid or structure alone are used jointly. Mostly this is done by exchanging information from time step to time step in an alternating fashion. These weak coupling methods are explicit and hence suffer from possible instabilities. Therefore often strong coupling––where equilibrium is satisfied jointly between fluid and structure in each time step––is desired; the simplest computational procedure is similar to the time stepping an alternating iteration. We show why also this approach may experience difficulties, and how they may be circumvented with block-Newton methods, still in the partitioned framework, by only using the solvers of the subproblems fluid and structure.

Partitioned but Strongly Coupled Iteration Schemes for Nonlinear Fluid-Structure Interaction

Abstract We look at the computational procedure of computing the response of a coupled fluid–structure interaction problem. We use the so-called strong fluid–structure coupling––a totally implicit formulation. At each time step in an implicit formulation, new values for the solution variables have to be computed by solving a nonlinear system of equations, where we assume that we have solvers for the subproblems. This is often the case, when we have existing software to solve each subproblem separately, and want to couple both. We show how to solve the overall nonlinear system by using only the solvers for the subproblems. This is achieved not by considering the equilibrium equations, but the fixed-point problem resulting from the solution iteration for each of the subproblems.

Finite elements for stochastic media problems

We consider discretization methods for stochastic partial differential equations, which are used to model media with stochastic properties. The Wick product formulation of the stochastic PDE is used, and it is shown how it may be discretized both spatially and in the stochastic part. A stochastic variational principle serves as the guideline for the discretization procedure. It is shown that both the projection onto the Wiener chaos and the Hermite-transform lead to the same fully discrete equations. The application examples show that there are other advantageous bases besides the Karhunen-Loeve expansion, and that the methods may be used in a variety of settings.

Application of hierarchical matrices for computing the Karhunen–Loève expansion

Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise. The uncertainties in the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. To approximate random fields with as few random variables as possible, but still retaining the essential information, the Karhunen–Loève expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of sparse hierarchical matrix techniques for this. A log-linear computational cost of the matrix-vector product and a log-linear storage requirement yield an efficient and fast discretisation of the random fields presented.

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.

Parameter identification in a probabilistic setting

Abstract The parameters to be identified are described as random variables, the randomness reflecting the uncertainty about the true values, allowing the incorporation of new information through Bayes’s theorem. Such a description has two constituents, the measurable function or random variable, and the probability measure. One group of methods updates the measure, the other group changes the function. We connect both with methods of spectral representation of stochastic problems, and introduce a computational procedure without any sampling which works completely deterministically, and is fast and reliable. Some examples we show have highly nonlinear and non-smooth behaviour and use non-Gaussian measures.

Nonlinear Galerkin methods for the model reduction of nonlinear dynamical systems

Numerical simulations of large nonlinear dynamical systems, especially over long-time intervals, may be computationally very expensive. Model reduction methods have been used in this context for a long time, usually projecting the dynamical system onto a sub-space of its phase space. Nonlinear Galerkin methods try to improve on this by projecting onto a sub-manifold which does not have to be flat. These methods are applied to the finite element model of a wind-turbine, where both the mechanical and the aerodynamical degrees of freedom can be considered for model reduction. For the internal forces (moments, section forces) the nonlinear Galerkin method gives a considerable increase in accuracy for very little computational cost.

Multiscale in time and stability analysis of operator split solution procedures applied to thermomechanical problems

Purpose - The purpose of this paper is to study the partitioned solution procedure for thermomechanical coupling, where each sub-problem is solved by a separate time integration scheme. Design/methodology/approach - In particular, the solution which guarantees that the coupling condition will preserve the stability of computations for the coupled problem is studied. The consideration is further generalized for the case where each sub-problem will possess its particular time scale which requires different time step to be selected for each sub-problem. Findings - Several numerical simulations are presented to illustrate very satisfying performance of the proposed solution procedure and confirm the theoretical speed-up of computations which follow from the adequate choice of the time step for each sub-problem. Originality/value - The paper confirms that one can make the most appropriate selection of the time step and carry out the separate computations for each sub-problem, and then enforce the coupling which will preserve the stability of computations with such an operator split procedure.

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.