Modeling of Single Crystal Magnetostriction Based on Numerical Energy Relaxation Techniques

In this work, an incremental energy minimization technique is proposed to simulate the magnetomechanically-coupled, nonlinear, anisotropic and hysteretic response of single crystalline magnetic shape memory alloys (MSMA). The model captures the three key physical mechanisms that cause this characteristic behavior, namely the field- or stress-induced martensite variant reorientation (twin boundary motion), magnetic domain wall motion, and local magnetization rotation, through an (incremental) energy minimizing evolution of internal state variables. Representative numerical response predictions are presented, compared to experimental observations, and discussed with respect to the associated microstructure evolution. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling the evolution of microstructures in finite plasticity

Kochmann and Hackl introduced in [1] a micromechanical model for finite single crystal plasticity. Based on thermodynamic variational principles this model leads to a non-convex variational problem. Employing the Lagrange functional, an incremental strategy was outlined to model the time-continuous evolution of a first order laminate microstructure. Although this model provides interesting results on the material point level, due to the global minimization in the evolution equations, the calculation time and numerical instabilities may cause problems when applying this model to macroscopic specimens. In order to avoid these problems, a smooth transition zone between the laminates is introduced to avoid global minimization, which makes the numerical calculations cumbersome compared to the model in [1]. By introducing a smooth viscous transition zone, the dissipation potential and its numerical treatment have to be adapted. We obtain rate-dependent time-evolution equations for the internal variables based on variational techniques and show as an example single slip shear. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A variational approach towards the modeling of equilibrium twin interfaces in a single-crystalline magnetic shape memory alloy sample

In this work, a variational approach is proposed to study the magneto-mechanical response of a single-crystalline MSMA sample. By proposing a total energy functional for the whole magneto-mechanical system, the governing PDE system is derived by calculating the variations of the total energy functional with respect to the independent variables. An iterative numerical algorithm is proposed to solve the governing PDE system. As an example, a MSMA sample with cuboid shape and subject to perpendicularly applied magnetic and mechanical loadings is considered. It can be seen that the magneto-mechanical response of this sample can be predicted at a quantitative level. The whole procedure of variant reorientation in the sample can also be simulated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Deformation Driven Homogenization of Fracturing Solids

The paper discusses numerical formulations of the homogenization for solids with discrete crack development. We focus on multi–phase microstructures of heterogeneous materials, where fracture occurs in the form of debonding mechanisms as well as matrix cracking. The definition of overall properties critically depends on the developing discontinuities. To this end, we extend continuous formulations [1] to microstructures with discontinuities [2]. The basic underlying structure is a canonical variational formulation in the fully nonlinear range based on incremental energy minimization. We develop algorithms for numerical homogenization of fracturing solids in a deformation–driven context with non–trivial formulations of boundary conditions for (i) linear deformation and (ii) uniform tractions. The overall response of composite materials with fracturing microstructures are investigated. As a key result, we show the significance of the proposed non–trivial formulation of a traction–type boundary condition in the deformation–driven context. (© 2005 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)     

Reverse Approximation of Energetic Solutions to Rate-Independent Processes

Energetic solutions to rate-independent processes are usually constructed via time-incremental minimization problems. In this work we show that all energetic solutions can be approximated by such incremental problems if we allow for approximate minimizers, where the error in minimization has to be of the order of the time step. Moreover, we study sequences of problems where the energy functionals have a Γ-limit. Research partially supported by Deutsche Forschungsgemeinschaft via the MATHEON project C18.     

Simulation of phase-transformations based on numerical minimization of intersecting Gibbs energy potentials

We present a novel approach for the simulation of solid to solid phase-transformations in polycrystalline materials. To facilitate the utilization of a non-affine micro-sphere formulation with volumetric-deviatoric split, we introduce Helmholtz free energy functions depending on volumetric and deviatoric strain measures for the underlying scalar-valued phase-transformation model. As an extension of affine micro-sphere models [5], the non-affine micro-sphere formulation with volumetric-deviatoric split allows to capture different Young's moduli and Poisson's ratios on the macro-scale [1]. As a consequence, the temperature-dependent free energy assigned to each individual phase takes the form of an elliptic paraboloid in volumetric-deviatoric strain space, where the energy landscape of the overall material is obtained from the contributions of the individual constituents. For the evolution of volume fractions, we use an approach based on statistical physics–taking into account actual Gibbs energy barriers and transformation probabilities [2]. The computation of individual energy barriers between the phases considered is enabled by numerical minimization of parametric intersection curves of elliptic Gibbs energy paraboloids. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A model for martensitic microstructure,its geometry and interface effects

In line with our formerly presented continuum micromechanical model [4,5] of the laminate microstructure of martensite, we here modify and reformulate the model to further its simplicity and clarity. The basic postulates are slightly altered. However, the approach remains the same, namely the kinematic assumption of nearly-planar laminate geometry put into an energy minimization framework with proper ansatzes of twin-interface and boundary energies. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Surface energies in microstructure of martensite

A continuum model for laminate geometry in microstructure of martensite, employing energy minimization, is presented. Ansatzes for twin interface energies as well as grain boundary energy are proposed. The model is applied to predict scale properties and geometrical characteristics of refinement and accommodation of microstructure. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularity of minimizers of non-isotropic integrals of the calculus of variations

The regularity of the minimizers of a special type of non-isotropic variational minimization problem is studied. The particularity of the potential of energy is that it has different growth rate with respect to different parts of the derivatives of the function. In particular, the model treated in this paper can be described as $$\Phi (Du) = |\partial _1 u|^2 + |\partial _2 u|^2 + |\partial _3 u|^2 + |\partial _3 u - |^p .$$ By using a result of P.Marcellini (cf. [4]) and perturbation method, it is proved that the minimizer of the Dirichlet boundary value problem is a function of W _loc ^{1, ∞} .     

On Variational Based Scale-Bridging of Inelastic Composites

The paper discusses formulations for the theoretical and numerical analysis of inelastic composites with scale separation. The basic underlying structure is a canonical variational setting in the fully nonlinear range based on incremental energy minimization. We focus on formulations of strain–driven homogenization for representative composite aggregates with emphasis on the development of canonical families of algorithms based on Lagrange and penalty functionals to cover alternative boundary constraints of (i.) linear deformations, (ii.) periodic deformations and (iii.) uniform tractions. As a key result, we present a compact matrix formulation for homogenization covering introduced alternative boundary constraints. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A variational arbitrary Lagrangian-Eulerian finite element formulation for standard dissipative solids at finite strains

A novel Arbitrary Lagrangian-Eulerian (ALE) finite element formulation for standard dissipative media at finite strains is presented. In contrast to previously published ALE approaches accounting for dissipative phenomena, the proposed scheme is fully variational. Consequently, no error estimates are necessary and thus, linearity of the problem and the corresponding Hilbert-space are not required. Hence, the resulting Variational Arbitrary Lagrangian-Eulerian (VALE) finite element method can be applied to highly nonlinear phenomena as well. In case of standard dissipative solids, so-called variational constitutive updates provide a variational principle. Based on these updates, the deformation mapping follows from minimizing an incrementally defined (pseudo) potential, i.e., energy minimization is the overriding criterion that governs every aspect of the system. Therefore, it is natural to allow the variational principle to drive mesh adaption as well. Thus, in the present paper, the discretizations of the deformed as well as the undeformed configuration are optimized jointly by minimizing the respective incremental energy of the considered mechanical system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The number of additional variables required for the integer programming formulation

It is known that a minimization problem having a finite feasible region with k elements can be formulated as an integer programming problem by introducing at most [log₂k] additional integer variables. In this note, we show that this bound is best possible in the sense that some minimization problem actually requires [log₂k] additional variables.     

Discussion on relationship between minimal energy and curve shapes

Energy minimization has been widely used for constructing curve and surface in the fields such as computer-aided geometric design, computer graphics. However, our testing examples show that energy minimization does not optimize the shape of the curve sometimes. This paper studies the relationship between minimizing strain energy and curve shapes, the study is carried out by constructing a cubic Hermite curve with satisfactory shape. The cubic Hermite curve interpolates the positions and tangent vectors of two given endpoints. Computer simulation technique has become one of the methods of scientific discovery, the study process is carried out by numerical computation and computer simulation technique. Our result shows that： （1） cubic Hermite curves cannot be constructed by solely minimizing the strain energy; （2） by adoption of a local minimum value of the strain energy, the shapes of cubic Hermite curves could be determined for about 60 percent of all cases, some of which have unsatisfactory shapes, however. Based on strain energy model and analysis, a new model is presented for constructing cubic Hermite curves with satisfactory shapes, which is a modification of strain energy model. The new model uses an explicit formula to compute the magnitudes of the two tangent vectors, and has the properties： （1） it is easy to compute; （2） it makes the cubic Hermite curves have satisfactory shapes while holding the good property of minimizing strain energy for some cases in curve construction. The comparison of the new model with the minimum strain energy model is included.     

Topology optimization methods for guided flow – comparison of optimality criteria vs. phase-field method

Optimization of guided flow problems is an important task for industrial applications especially those with high Reynolds numbers. There exist several optimization methods to increase the energy efficiency of these problems. Different optimization methods are shown bei Klimetzek [1], Hinterberger [2] and Pingen [3]. In recent years the phase-field method has been shown to be an applicable method for different kinds of topology optimization [4, 5]. We present results of topology optimization methods with optimality criterion and by using a phase-field model in the area of guided fluid flow problems. The two methods aim on the same main target reducing the pressure drop between the inlet and outlet of the flow domain. The first method is based on local optimality criterion, preventing the backflow in the flow domain [1, 6, 7]. The second method is based on a phase field model, which describes a minimization problem and uses a specially constructed driving force to minimize the total energy of the system [4, 5]. We investigate the capabilities and limits of both methods and present examples of different resulting geometries. The initial configurations are prepared in a way that the same optimization problem is solved with both methods. We discuss these results regarding the shape of the improved flow geometry. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A newton-penalty method for a simplified liquid crystal model

In this paper we are concerned with the computation of a liquid crystal model defined by a simplified Oseen-Frank energy functional and a (sphere) nonlinear constraint. A particular case of this model defines the well known harmonic maps. We design a new iterative method for solving such a minimization problem with the nonlinear constraint. The main ideas are to linearize the nonlinear constraint by Newton’s method and to define a suitable penalty functional associated with the original minimization problem. It is shown that the solution sequence of the new minimization problems with the linear constraints converges to the desired solutions provided that the penalty parameters are chosen by a suitable rule. Numerical results confirm the efficiency of the new method.     

Comparison of a mixed least-squares formulation using different approximation spaces

The main goal of the present work is the comparison of the performance of a least-squares mixed finite element formulation where the solution variables (displacements and stresses) are interpolated using different approximation spaces. Basis for the formulation is a weak form resulting from the minimization of a least-squares functional, compare e.g. [1]. As suitable functions for standard interpolation polynomials of Lagrangian type are chosen. For the conforming discretization of the Sobolev space vector-valued Raviart-Thomas interpolation functions, see also [2], are used. The resulting elements are named as P_mP_k and RT_mP_k. Here m (stresses) and k (displacements) denote the approximation order of the particular interpolation function. For the comparison we consider a two-dimensional cantilever beam under plain strain conditions and small strain assumptions. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Incremental proximal methods for large scale convex optimization

We consider the minimization of a sum \({\sum_{i=1}^mf_i(x)}\) consisting of a large number of convex component functions f _i . For this problem, incremental methods consisting of gradient or subgradient iterations applied to single components have proved very effective. We propose new incremental methods, consisting of proximal iterations applied to single components, as well as combinations of gradient, subgradient, and proximal iterations. We provide a convergence and rate of convergence analysis of a variety of such methods, including some that involve randomization in the selection of components. We also discuss applications in a few contexts, including signal processing and inference/machine learning.     

Unchained polygons and the N-body problem

We study both theoretically and numerically the Lyapunov families which bifurcate in the vertical direction from a horizontal relative equilibrium in ℝ³. As explained in [1], very symmetric relative equilibria thus give rise to some recently studied classes of periodic solutions. We discuss the possibility of continuing these families globally as action minimizers in a rotating frame where they become periodic solutions with particular symmetries. A first step is to give estimates on intervals of the frame rotation frequency over which the relative equilibrium is the sole absolute action minimizer: this is done by generalizing to an arbitrary relative equilibrium the method used in [2] by V. Batutello and S. Terracini. In the second part, we focus on the relative equilibrium of the equal-mass regular N-gon. The proof of the local existence of the vertical Lyapunov families relies on the fact that the restriction to the corresponding directions of the quadratic part of the energy is positive definite. We compute the symmetry groups G _r/s (N, k, η) of the vertical Lyapunov families observed in appropriate rotating frames, and use them for continuing the families globally. The paradigmatic examples are the “Eight” families for an odd number of bodies and the “Hip- Hop” families for an even number. The first ones generalize Marchal’s P ₁₂ family for 3 bodies, which starts with the equilateral triangle and ends with the Eight [1, 3–6]; the second ones generalize the Hip-Hop family for 4 bodies, which starts from the square and ends with the Hip-Hop [1, 7, 8]. We argue that it is precisely for these two families that global minimization may be used. In the other cases, obstructions to the method come from isomorphisms between the symmetries of different families; this is the case for the so-called “chain” choreographies (see [6]), where only a local minimization property is true (except for N = 3). Another interesting feature of these chains is the deciding role played by the parity, in particular through the value of the angular momentum. For the Lyapunov families bifurcating from the regular N-gon whith N ≤ 6 we check in an appendix that locally the torsion is not zero, which justifies taking the rotation of the frame as a parameter. To the memory of J. Moser, with admiration