Fluid-Structure Interactions in ALE and Fully Eulerian Coordinates

We present a novel approach to treat fluid-structure interactions in a closed variational setting. The standard model is the so-called ‘arbitrary Lagrangian-Eulerian’ (ALE) framework. Here, the fluid is transformed into an artificial coordinate system, which fits with the Lagrangian structure system. In the fully Eulerian framework, which is the reverse to ALE, the fluid is left in its natural coordinates while structure is transformed. With this approach, very large deformations, change of topology as well as free movement of the structure can be computed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On The Formulatio and Numerical Implementation of Dissipative Electro-Mechanics at Large Strains

In the last year an increasing interest in functional materials such as ferroelectric polymers and ceramics has been shown. For those materials viscous effects or electric polarizations cause hysteresis phenomena accompanied with possibly large remanent strains. This paper outlines aspects of the formulation and numerical implementation of dissipative electro-mechanics at large strains. In the first part, we focus on the geometric nature of dissipative electro-mechanics. In a second part, we discuss constitutive assumptions which account for specific problems arising in the geometric nonlinear setting. This concerns the definition of objective energy storage and dissipation functions with suitable symmetries and (weak) convexity properties. With regard to the choice of the internal variables entering these functions, a critical point are kinematic assumptions. Here, we investigate the multiplicative decomposition of the local deformation gradient into reversible and remanent parts as well as the introduction of a remanent metric. In a third part, we summarize details of the constitutive updates as well as the finite element formulations, both on the basis of compact incremental variational principles. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence of a finite element/ALE method for the Stokes equations in a domain depending on time

We consider the approximation of the unsteady Stokes equations in a time dependent domain when the motion of the domain is given. More precisely, we apply the finite element method to an Arbitrary Lagrangian Eulerian (ALE) formulation of the system. Our main results state the convergence of the solutions of the semi-discretized (with respect to the space variable) and of the fully-discrete problems towards the solutions of the Stokes system.     

A fully variational algorithmic formulation for wrinkling at finite strains

This contribution is concerned with an efficient novel algorithmic formulation for wrinkling at finite strains. In contrast to previously published numerical implementations, the advocated method is fully variational. More precisely, the parameters describing wrinkles or slacks, together with the unknown deformation mapping, are computed jointly by minimizing the potential energy of the considered mechanical system. Furthermore, the wrinkling criteria are naturally included within the presented variational framework. The presented approach allows to employ three–dimensional constitutive models directly, i.e., plane stress conditions characterizing membranes are variationally enforced by minimizing the potential energy with respect to the transversal strains. Since the proposed formulation for wrinkling in membranes is fully variational, it can be conveniently combined with other variational methods (based on energy minimization). As an example, a variationally consistent framework for finite strain plasticity theory is considered. More precisely, the minimization principle characterizing wrinkling in elastic membranes and that describing plasticity in inelastic solids are coupled leading to a novel variational approach for inelastic membranes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Towards variational constitutive updates for finite strain plasticity models based on non-associative evolution equations

In this contribution, first steps towards variational constitutive updates for finite strain plasticity theory based on non-associative evolution equations are presented. These schemes allow to compute the unknown state variables such as the plastic part of the deformation gradient, together with the deformation mapping, by means of a fully variational minimization principle. Therefore, standard optimization algorithms can be applied to the numerical implementation leading to a very robust and efficient numerical implementation. Particularly, for highly non-linear, singular or nearly ill-posed physical models like that corresponding to crystal plasticity showing a large number of possible active slip planes, this is a significant advantage compared to standard constitutive updates such as the by now classical return-mapping algorithm. While variational constitutive updates have been successfully derived for associative plasticity models, their extension to more complex constitutive laws, particularly to those featuring non-associative evolution equations, is highly challenging. In the present contribution, a certain class of non-associative finite strain plasticity models is discussed and recast into a variationally consistent format. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of steady and unsteady flow over a vibrating airfoil in a channel

The work deals with a numerical solution of 2D steady and unsteady inviscid incompressible flow over the profile NACA 0012 in a channel. The finite volume method (FVM) in a form of cell-centered explicit schemes at quadrilateral C-mesh is used. Governing system of equations is the system of incompressible Euler equations. The method of artificial compressibility and time dependent method is applied to steady computations. The small disturbance theory (SDT) applied to a numerical solution of flow over a rotated profile by a small angle only is mentioned. Brief introduction is given to the Arbitrary (Semi) Lagrangian-Eulerian (ALE) method used for unsteady computations. Some numerical results of unsteady flow over a vibrating profile achieved by both SDT and ALE method are presented. Unsteady flow is caused by prescribed oscillations of the profile (one degree of freedom) fixed in an elastic axis. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An evolution equation based approach to topology optimization

The objective of topology optimization is to find a mechanical structure with maximum stiffness and minimal amount of used material for given boundary conditions [2]. There are different approaches. Either the structure mass is held constant and the structure stiffness is increased or the amount of used material is constantly reduced while specific conditions are fulfilled. In contrast, we focus on the growth of a optimal structure from a void model space and solve this problem by introducing a variational problem considering the spatial distribution of structure mass (or density field) as variable [3]. By minimizing the Gibbs free energy according to Hamilton's principle in dynamics for dissipative processes, we are able to find an evolution equation for the internal variable describing the density field. Hence, our approach belongs to the growth strategies used for topology optimization. We introduce a Lagrange multiplier to control the total mass within the model space [1]. Thus, the numerical solution can be provided in a single finite element environment as known from material modeling. A regularization with a discontinuous Galerkin approach for the density field enables us to suppress the well-known checkerboarding phenomena while evaluating the evolution equation within each finite element separately [4]. Therefore, the density field is no additional field unknown but a Gauß-point quantity and the calculation effort is strongly reduced. Finally, we present solutions of optimized structures for different boundary problems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fully variational strong discontinuity approach at finite strains

A novel, fully variational three-dimensional finite element formulation for the modeling of locally embedded strong discontinuities at finite strains is presented. The proposed numerical model is based on the Enhanced Assumed Strain concept with an additive decomposition of the displacement gradient into a conforming and an enhanced part. The discontinuous component of the displacement field which is associated with the failure in the modeled structure is isolated in the enhanced part of the deformation gradient. In contrast to previous works, a variational constitutive update is used. The internal variables are determined by minimizing a pseudo-elastic potential. The advantages of such a formulation are well known, e.g. the tangent stiffness matrix is symmetric, standard optimization algorithms can be applied and it represents a natural basis for error estimation and mesh adaption. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reduced-dissipation remapping of velocity in staggered arbitrary Lagrangian-Eulerian methods

Remapping is an essential part of most Arbitrary Lagrangian-Eulerian (ALE) methods. In this paper, we focus on the part of the remapping algorithm that performs the interpolation of the fluid velocity field from the Lagrangian to the rezoned computational mesh in the context of a staggered discretization. Standard remapping algorithms generate a discrepancy between the remapped kinetic energy, and the kinetic energy that is obtained from the remapped nodal velocities which conserves momentum. In most ALE codes, this discrepancy is redistributed to the internal energy of adjacent computational cells which allows for the conservation of total energy. This approach can introduce oscillations in the internal energy field, which may not be acceptable. We analyze the approach introduced in Bailey (1984) [11] which is not supposed to introduce dissipation. On a simple example, we demonstrate a situation in which this approach fails. A modification of this approach is described, which eliminates (when it is possible) or reduces the energy discrepancy.     

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems is proposed based on the extended Hamiltonian principle, the Hamilton-Jacobi-Bellman (HJB) equation and its variational integral equation, and the finite time element approximation. The differential extended Hamiltonian equations for structural vibration systems are replaced by the variational integral equation, which can preserve intrinsic system structure. The optimal control law dependent on the value function is determined by the HJB equation so as to satisfy the overall optimality principle. The partial differential equation for the value function is converted into the integral equation with variational weighting. Then the successive solution of optimal control with system state is designed. The two variational integral equations are applied to sequential time elements and transformed into the algebraic equations by using the finite time element approximation. The direct optimal control on each time element is obtained respectively by solving the algebraic equations, which is unconstrained by the system state observed. The proposed control algorithm is applicable to linear and nonlinear systems with the quadratic performance index, and takes into account the effects of external excitations measured on control. Numerical examples are given to illustrate the optimal control effectiveness.     

Advanced Solution Strategies for r-Adaptive Finite Element Analysis

In Finite Element Analysis (FEA) the discretisation has wide influence on the quality of the analysis. With r-adaptive FEA it is aimed to improve the finite element solution by finding the optimal mesh without changing the mesh connectivity and the order of the elements. Thus, this approach belongs to the group of mesh-moving methods. The r-adaptivity approach presented is governed by energy minimisation and therefore is called energy-based. It is built upon a variational Arbitrary Lagrangian-Eulerian (vALE) formulation whereby the potential energy is varied with respect to spatial and material coordinates. However, even for simple problems the Hessian is likely to be singular or indefinite. This complicates the application of solution schemes based on Newton's method. Motivated by the approaches of [1–4], we try to find appropriate numeric methods for r-adaptivity. For this purpose, we study the numerical performance of a primal barrier scheme, of an augmented Lagrange barrier scheme and the primal-dual interior point package IPOPT. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formulation and survey of ALE method in nonlinear solid mechanics

This paper investigates the applicability and accuracy of existing formulation methods in general purpose finite element programs to the finite strain deformation problems. The basic shortcomings in using such programs in these applications are then pointed out and the need for a different type of formulation is discussed. An arbitrary Lagrangian-Eulerian (ALE) method is proposed and a concise survey of ALE formulation is given. A consistent and complete ALE formulation is derived from the virtual work equation transformed to arbitrary computational reference configurations. Differences between the proposed formulations and similar ones in the literature are discussed. The proposed formulation presents a general approach to ALE method. It includes load correction terms and is suitable for rate-dependent and rate-independent material constitutive law. The proposed formulation reduces to both updated Lagrangian and Eulerian formulations as special cases.     

On the effect of different parameterizations of the flow rule on the efficiency of variational constitutive updates

The numerical efficiency of so-called variational constitutive updates for finite strain plasticity theory is analyzed. These updates compute the unknowns such as the plastic strains by minimizing an appropriate functional. Within the present paper, different parameterizations of the flow rule are utilized within the variational constitutive update scheme. It is shown that comparing to the return-mapping algorithm, the variational updates require significantly less iteration steps and thus, is numerically highly efficient. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A novel h-adaptive finite element method for standard dissipative media at finite strains based on energy minimization

A novel h-adaptive finite element strategy for standard dissipative media at finite strains based on energy minimization is presented. The method can be applied to any (incremental) minimization problem to be analyzed by finite elements. Similarly to an error estimator by Babǔska & Rheinboldt , the proposed error indicator is based on solving a local Dirichlet -type problem. However, in contrast to the original work, a different error indicator is considered. Provided the underlying physical problem is governed by a minimization problem, the difference between the energy of the elements defining the local problem computed from the initial finite element interpolation and that associated with the local Dirichlet -type problem is used as an indicator. If this difference reaches a certain threshold, the elements defining the local problem are refined by applying a modified longest edge bisection according to Rivara . Since this re-meshing strategy leads to a nested family of triangulations, the transfer of history variables necessary to describe dissipative materials is relatively inexpensive. The presented h-adaption is only driven by energy-minimization. As a consequence, anisotropic meshes may evolve if they are energetically favorable. The versatility and rate of convergence of the resulting approach are illustrated by means of selected numerical examples. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Advanced finite element formulations for modeling thin piezoelectric structures

This contribution is concerned with mixed finite element formulations for modeling piezoelectric beam and shell structures. Due to the electromechanical coupling, specific deformation modes are joined with electric field components. In bending dominated problems incompatible approximation functions of these fields cause incorrect results. These effects occur in standard finite element formulations, where interpolation functions of lowest order are used. A mixed variational approach is introduced to overcome these problems. The mixed formulation allows for a consistent approximation of the electromechanical coupled problem. It utilizes six independent fields and could be derived from a Hu-Washizu variational principle. Displacements, rotations and the electric potential are employed as nodal degrees of freedom. According to the Timoshenko theory (beam) and the Reissner-Mindlin theory (shell), the formulations account for constant transversal shear strains. To incorporate three dimensional constitutive relations all transversal components of the electric field and the strain field are enriched by mixed finite element interpolations. Thus the complete piezoelectric coupling is appropriately captured. The common assumption of vanishing transversal stress and dielectric displacement components is enforced in an integral sense. Some numerical examples will demonstrate the capability of the presented finite element formulation. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixed finite element analysis for dissipative SRLW equations with damping term

In this paper a mixed finite element (MFE) formulation is proposed for the initial-boundary value problem of dissipative symmetric regularized long wave (SRLW) equations with damping. Existence and uniqueness of its generalized solution and of the fully discrete mixed finite element solution are proved. Error estimates based on energy methods are given. Numerical experiments verify the theoretical analysis.     

Hybrid stress finite element formulation based on additional internal displacements – application of microscopic geometric perturbation and extension to linear dynamics

Based on the variational principle for hybrid stress finite element method, the assumed stress distribution can be determined through a variational process which includes a microscopic geometric perturbation. An interpretation of such geometric perturbation is discussed. For the creation of, what is called, hyper-bioelement and for the purpose of maintaining invariance, tensor notations of the physical quantities are represented by the use of covariant and contravariant basis vectors of the natural coordinate system. This process is illustrated by a simple 4-node plane element. Furthermore, based on the legitimate variational principle for linear dynamics and the presently proposed method to choose the assumed stress distribution, the finite element formulation different from the ordinary formula is advocated.