• Associated people: 
  A. Schwarz, K.Steeger, J. Schröder, G. Starke (Leibniz University Hannover)

  
  

• Abstract:
  For the solution of partial differential equations describing physical problems
  in the framework of solid mechanics,
  we develop in this project least-squares mixed
  finite element formulations and examine their performance.

  Starting from the balance of momentum, we obtain div-grad first-order systems
  with e.g. stresses and displacements as unknowns and by means of
  quadratic L₂-norms a least-squares functional can be constructed.
  This functional is the basis for
  the associated minimization problem, which is not restricted to the
  LBB-condition.  

  Moreover, due to some additional advantages, as e.g.
  a smooth stress approximation even for quasi-incompressible materials and
  an a posteriori error estimator without additional costs,
  least-squares variational principles have increasingly gained attention.

• References
  Schwarz, A., Schröder, J. & Starke, G. [2010], “A modified Least-Squares Mixed Finite Element with improved Momentum Balance”, International Journal for Numerical Methods in Engineering 81, p. 286-306.

  Schwarz, A. [2009], “Least-Squares Mixed Finite Elements for Solid Mechanics”, Dissertation, Institut für Mechanik, Bericht 7, Universität Duisburg-Essen.

  Schwarz, A., Schröder, J. & Starke, G. [2009], “Least-Squares Mixed Finite Elements for Small Strain Elasto-Viscoplasticity”, International Journal for Numerical Methods in Engineering 77, p. 1351-1370.

• Associated people: 
  A. Schwarz, J. Schröder

  
  

• Abstract:
  The main focus of this project is to examine least-squares mixed finite elements for the solution of steady and unsteady Newtonian fluid flow, which is described by the incompressible Navier-Stokes equations.
  Least-squares variational principles have increasingly gained attention, which is due to some theoretical and computational advantages compared to the Galerkin method.
  • The method provides for instance an a posteriori error estimator without additional costs, which can be used for the development of adaptive mesh refinement algorithms.
  
  
  • The resulting symmetric positive definite system matrices can be solved by using robust and fast iterative methods even for problems with governing nonselfadjoint operators such as fluid dynamics and transport problems.
  
  
  • The inf-sup condition does not hold, so there are no restrictions for the choice of the polynomial degree of the finite element spaces.
  
  The development and improvement of mixed least-squares finite element formulations by means of these topics are the main incentive of our research.
  In general, we consider div-grad first-order systems resulting in approaches with e.g. stresses, velocities, and/or pressure as unknowns. The L₂-norms of the residuals of the derived equations yield then the least-squares functional, which is the basis for the associated minimization problem.
• References
  Cai, Z. and Lee, B. and Wang, P.[2004], “Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems”, SIAM Journal on Numerical Analysis 42, p. 843-859.

  Schwarz, A. and Schröder, J. [submitted], “A mixed least-squares formulation of the Navier-Stokes equations for incompressible Newtonian fluid flow”, PAMM Proceedings in Applied Mathematics and Mechanics.
   
  Münzenmaier, S. and Starke,G. [2011], “First-order system least squares for coupled Stokes-Darcy flow”, SIAM Journal on Numerical Analysis 49, p. 387--404.

• Associated people:
  J. Schröder, D. Balzani, D. Brands

• Abstract:
  Micro-heterogeneous high-tech steels enable ambitious engineering constructions. For the design of such random multi-phase microstructures numerical simulation procedures are important for the reliable prediction of the overall response of macroscopic boundary value problems. Direct homogenization schemes, which are designed for such kind of problems, can only be efficient if the complex random microstructure is approximated by significantly less complex representative volume elements (RVEs) capturing the basic morphological attributes of the microscale. This is the main task in the project, where several statistical measures (spectral-density, lineal-path function, Minkowski valuations, etc.) are applied for the construction of statistically similar RVEs (SSRVEs). A main challenge in this period is the design of three-dimensional SSRVEs based on the Electron-Backscatter Diffraction (EBSD) combined with Focused-Ion-Beam (FIB) measurements. The performance of the direct homogenization approach using the constructed SSRVEs is analyzed for representative macroscopic deep-drawing processes.
• References:
  Balzani, D., Brands, D., Schröder, J., and Carstensen, C. [2010], "Sensitivity Analysis of Statistical Measures for the Reconstruction of Microstructures based on the Minimization of Generalized Least-Square Functionals", Technische Mechanik, 30, 297-315

  Schröder, J., Balzani, D., and Brands, D. [2010], "Approximation of Random Microstructures by Periodic Statistically Similar Representative Volume Elements based on Lineal-Path Functions", Archive of Applied Mechanics

• Associated people:
  J. Schröder, D. Balzani, D. Brands

• Abstract:
  Advanced high strength steels are suitable materials to optimize stability while reducing weight. During the process of manufacture the material exhibits large plastic deformations. Since the distinct microstructure is a crucial reason for the material properties, its incorporation during the numerical analysis is necessary. A numerical tool for the direct incorporation of micromechanical information is the FE2-method, also known as direct micro-macro-transition procedure. Thereby in each macroscopic integration point a boundary value problem is solved on the microscale, where a representative volume element covers the typical morphology of the considered material.

• Associated people:
  J. Schröder, P. Neff, V. Ebbing

• Abstract:
  In order to guarantee the existence of minimizers of the underlying potential in finite elasticity, a suitable concept is the polyconvexity condition of the energy function in the sense of Ball 1977. Until recently, the construction of polyconvex energy functions for the description of anisotropic material behavior was a question yet to be answered. The first transversely isotropic and orthotropic polyconvex energy functions were presented by Schröder and Neff in the years 2001, 2003. In the present work a method for the construction of polyconvex, coordinate-invariant energy functions for the description of all existing anisotropic hyperelastic materials is proposed. The key idea is the introduction of so-called crystallographically motivated structural tensors. The applicability of this method is shown within several numerical examples.
• References:
  Ebbing, V. [2010], "Design of Polyconvex Energy Functions for All Anisotropy Classes", Dissertation, Institut für Mechanik, Bericht Nr. 8, Universität Duisburg-Essen

  Ebbing, V., Schröder, J., Neff, P. [2010], "Construction of Polyconvex Energies for Non-Trivial Anisotropy Classes", in Poly-, Quasi- and Rank-One Convexity in Applied Mechanics, CISM Courses and Lectures, Editors: J. Schröder and P. Neff, Springer, Vol. 516, 107-130

  Schröder, J., Neff, P., Ebbing, V. [2008], "Anisotropic Polyconvex Energies on the Basis of Crystallographic Motivated Structural Tensors", Journal of the Mechanics and Physics of Solids, Vol. 56, 3486-3506
  

• Associated people:
  J. Schröder, M.-A. Keip

• Abstract:
  Electro-mechanically coupled solids as piezo- and ferroelectric ceramics are in general polycrystalline materials and possess a heterogeneous micro- and mesostructure. Therefore, the overall macroscopic behavior of a bulk ferroelectric is influenced by the physical and geometrical constitution of the underlying micro- and mesostructure. A bulk macroscopic ferroelectric is generally composed of grains (polycrystalline ferroelectrics) or artificially constructed mesostructures that are composed of at least one piezoelectric phase (piezocomposites). In each case, the overall macroscopic behavior as well as the individual fields on the lower scales are of special interest, since both are important for the functionality of the piezoactive ceramic.

  The project deals with the two-scale simulation of electro-mechanically coupled materials by means of the FE^2-method. In this connection, the focus is on the computation of effective material parameters and the analysis of mesoscopic electro-mechanical fields.
  

• References:
  J. Schröder & M.-A. Keip (2011). Multiscale modeling of electro-mechanically coupled materials: Homogenization procedure and computation of overall moduli. In: M. Kuna, A. Ricoeur (eds.) IUTAM Symposium on Multiscale Modelling of Fatigue, Damage and Fracture in Smart Materials, IUTAM Bookseries, Vol. 24, pp. 265-276, Springer Netherlands

  J. Schröder & M.-A. Keip (2010). A framework for the two-scale homogenization of electro-mechanically coupled boundary value problems. In: M. Kuczma, K. Wilmanski (eds.) Computer Methods in Mechanics, Advanced Structured Materials, Vol. 1, pp. 311-329, Springer Berlin Heidelberg

• Associated people:
  J. Schröder, M.-A. Keip

• Abstract:
  When applied to high electrical and/or mechanical fields, the constitutive response of ferroelectric materials becomes highly nonlinear and expresses itself in terms of e.g. dielectric-, butterfly-, and ferroelastic-type hysteresis. This is due to the fact that high electro-mechanical fields induce internal switching of the existing spontaneous polarization within the material. Such switching is utilized during the plarization process of a polycrystalline ferroelectric in order to produce a macroscopically coupled functional electroceramic. On the other hand, during the operation of a poled piezoelectric device, high internal fields can arise and lead to electrically and mechanically induced switching. These fields influence the functionality of the ceramic in such a way that they can lead to a partial or complete depolarization of the material.

  The research project deals with the continuum mechanical modeling of electrically and mechanically induced switching from a microscopic view point. The modeling is based on a fully tetragonal representation of the material including an energy-based concept for the incorporation of ferroelectric and ferroelastic switching.
  

• References:
  M.-A. Keip & J. Schröder (2010). A tetragonal switching model for ferroelectric materials. Proceedings in Applied Mathematics and Mechanics 10(1): 369-370
  

• Associated people:
  J. Schröder, A. Klawonn, D. Balzani, O. Rheinbach, D. Brands, S. Brinkhues

• Abstract:
  In order to improve the blood flow in artherosclerotic arteries an
  extension of the lumen is essential. An often used method of treatment
  is the balloon-angioplasty, whose success requires the knowledge of the
  mechanics of arterial walls. Thus the numerical simulation provides
  a method to analyse the deformation behavior of the arterial wall
  and its constituents under the loading by a ballon-angioplasty.
  The necessary 3D-discretizations by finite elements results in large
  systems of equations, therefore, a parallel algorithm using FETI-DP is
  applied to solve the equilibrium problem.
  

 

• References:
  Balzani, D., Brands, D., Klawonn, A., Rheinbach, O. & Schröder, J. (2010), "On the mechanical modeling of anisotropic biological soft tissue and iterative parallel solution strategies", Archive of Applied Mechanics. Vol. 80, pp. 479-488.

  Brands, D., Klawonn, A., Rheinbach, O. & Schröder, J. (2008), "Modelling and convergence in arterial wall simulations using a parallel FETI solution strategy", Computer Methods in Biomechanics and Biomedical Engineering. Vol. 11, pp. 569-583.

• Associated people:
  J. Bluhm, T. Ricken, W. M. Bloßfeld

• Abstract:
  In civil engineering, the processes of freezing and thawing of fluid and gas saturated porous media is a point of great discussion. In addition they are strongly influenced by the fluid-ice phase transition. Ice formation in porous media results from coupled heat and mass transport and is accompanied by ice expansion. In a multiphase approach, a macroscopic quadruple model consisting of the constituents solid (cement stone), liquid (freezable water), ice and gas is developed within the framework of the Theory of Porous Media (TPM). The usefulness of the model for saturated porous solids under cycling thermal loading is demonstrated by a comparison of computationally and experimentally gained data of the CIF-Test (Capillary suction, Internal damage and Freeze-Thaw Test). Beside the influence of the heat of fusion, especially, the capillary suction as well as frost suction in gas and liquid filled porous media will be studied.

• References:
  Ricken, T & Bluhm, J. [2010], "Modeling fluid saturated porous media under frost attack", GAMM-Mitt., Vol. 33 (1), pp. 40-56.

  Bluhm, J., Ricken, T., & Blossfeld, M. [2009], "Dynamic Phase Transition Border under Freezing-Thawing Load in Porous Media -- A Multiphase Description", Schroeder, J. (ed.): Report 47, Institute of Mechanics, University Duisburg-Essen, Germany.

  Bluhm, J. and Ricken, T. & Blossfeld, M. [2009], "Freezing and thawing load in porous media - Experiment and Simulation", PAMM, Vol. 9, pp. 387 -388.

  Bluhm, J. & Ricken, T. & Blossfeld, M. [2008], "Energetische Aspekte zum Gefrierverhalten von Wasser in porösen Strukturen", PAMM, Vol. 8, pp. 10483 - 10484.

Associated people:
B. Eidel, J. Schröder

Abstract:
Pyramidal microindentation into the (001) surface of an fcc single crystal has shown indent shapes which strongly depend on the azimuthal orientation of the pyramid, see the left column of the adjacent figure. This observation is experimentally elucidated by means of high resolution electron back-scatter diffraction (EBSD) technique along with digital image processing creating a digital surface model. The main findings are that pile-up formation is invariantly maximum in <110> directions (4 hillocks emerge) thus being independent of the azimuthal orientation of the pyramid. For different orientations of the indenter the material pile-up is locally accommodated to the indenter faces leading to a convex, a concave or a mixed curved contact rim at the faces of the indenter. The right column in the figure shows, that crystal plasticity finite element simulations are in good agreement with the observed surface deformation pattern. The influence of stress concentration onto the anisotropic pile-up is negligible. This is corroborated on the relative invariance of pile-up for different indenter orientations (an anisotropy in loading). The driving mechanism behind the observed phenomena is plastic glide in (111) <110> slip systems. 

 

References

Eidel, B. (2011), "Crystal plasticity finite-element analysis versus experimental results of
pyramidal indentation into (001) fcc single crystal", Acta Materialia. 59, pp. 1761-1771.

Eidel, B. & Gruttmann, F. (2007), "Squaring the circle - A curious phenomenon of fcc single
crystals in spherical microindentation", Computational Materials Science. Vol. 39(1), pp. 172-178.

Eidel, B. (2004), "Anisotropic Inelasticity - Modelling, Simulation, Validation" , pp. 230.
PhD Thesis, TU-Darmstadt, Shaker-Verlag.

Associated people:
B. Eidel, C. Kuhn (LTM, Uni Kaiserslautern), J. Schröder

Abstract:
Time integration is the numerical kernel of inelastic finite element calculations, which largely determines their accuracy and efficiency. If higher order Runge–Kutta (RK) methods, p>2, are used for integration in a standard manner, they do not achieve full convergence order but fall back to second-order convergence. This deficiency called order reduction is a longstanding problem in computational inelasticity. We analyze it for viscoelasticity, where the evolution equations follow ordinary differential equations (ODE). We focus on RK methods of third order. We prove that the reason for order reduction is the (standard) linear interpolation of strain to construct data at the RK-stages within the considered time interval. We prove that quadratic interpolation of strain based on t_(n), t_(n+1) and, additionally, t_(n-1) data implies consistency order three for total strain, viscoelastic strain and stress. Simulations applying the novel interpolation technique are in perfect agreement with the theoretical predictions. The present methodology is advantageous, since it preserves the common, staggered structure of finite element codes for inelastic stress calculation. Furthermore, it is easy to implement, the overhead of additional history data is small and the computation time to obtain a defined accuracy is considerably reduced compared with backward Euler. 

Ongoing research work has been focused on the more delicate problem of order reduction in elasto-plasticity. This broad class of inelastic continuum constitutive laws is usually described by ordinary differential equations for the evolution of plastic flow, which is subject to the yield condition as an algebraic constraint, thus forming altogether a set of differential algebraic equations (DAE).

 

References

Eidel, B. & Kuhn, C. (2011), "Order reduction in computational inelasticity: why it happens and how to overcome it - the ODE-case of viscoelasticity", International Journal for Numerical Methods in Engineering. Vol. accepted for publication.

 

Associated people:
B. Eidel, A. Stukowski (LLNL, USA), J. Schröder

Abstract:  The quasi-continuum (QC) method is a prominent example of a bottom-up, concurrent multiscale method aiming at a seamless link of atomic with continuum length scales. This aim is achieved by three main building blocks (i) a coarse-graining of fully atomic resolution via finite element discretization in order to reduce the number of degrees of freedom. Fully atomic resolution is retained at hot spots of inelastic deformations like at crack tips, at defect cores or alike, whereas a finite element coarse-graining is applied in regions of purely elastic deformation. (ii) An approximation of the energy/forces in coarse-grained regions via numerical quadrature which avoids the explicit computation of the site energy of each and every atom. (iii) Adaptivity, i.e. spatially adaptive resolution, is necessary to provide full atomic resolutions in regions of evolving or moving inelastic deformations like in crack propagation or microstructural evolution.                                                                       

In [Eidel & Stukowski 2009] we develop a novel QC approach aiming at a truly seamless transition from the atomic to the continuum description of crystalline solids at zero temperature. It heavily draws on the framework proposed by Knap and Ortiz (2001). Opposed to Knap and Ortiz, the energy instead of forces is subject to a cluster based sampling scheme with adaptive resolution. We show that only the present ansatz endows the QC theory with a variational structure. The fully nonlocal methodology is assessed in nanoindentation into an fcc single crystal. Compared to the fully atomistic counterpart of lattice statics, the coarse-grained atomistic description with adaptive resolution achieves good agreement with respect to the force-displacement curve, the load-level and locus of dislocation nucleation and the dislocation microstructure for a small fraction of the computational costs. The figure shows the dislocation microstructure below the surface of an fcc aluminum single crystal which is subject to ball indentation at the nanoscale.    

More recent research has been concerned with free surface relaxations of nano-structures in collaboration with N.V. Prajapati (RUB). Ongoing research activities are focused on modeling extensions of the QC but also with the development of novel numerical features for the method.

 

References

Eidel, B. & Stukowski, A. (2009), "A variational formulation of the quasicontinuum method based on energy sampling in clusters", Journal of the Mechanics and Physics of Solids. Vol. 57(1), pp. 87-108.

Eidel, B. (2009), "Coupling atomistic accuracy with continuum effectivity for predictive simulations in materials research - the Quasicontinuum Method", International Journal of Materials Research. Vol. 100(11), pp. 1503-1512.