First-Order System Least Squares and the Energetic Variational Approach for Two-Phase Flow

This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-phase flow. The main goal is to show that this discretization, along with numerical techniques such as nested iteration, algebraic multigrid, and adaptive local refinement, can be used to solve these types of complex fluid flow problems. In addition, from an energetic variational approach, it can be shown that an important quantity to preserve in a given simulation is the energy law. We discuss the energy law and inherent structure for two-phase flow using the Allen-Cahn interface model and indicate how it is related to other complex fluid models, such as magnetohydrodynamics. Finally, we show that, using the FOSLS framework, one can still satisfy the appropriate energy law globally while using well-known numerical techniques.Comment: 22 pages, 8 figures submitted to Journal of Computational Physic

Mixed Galerkin and least-squares formulations for isogeometric analysis.

This work is concerned with the use of isogeometric analysis based on Non- Uniform Rational B-Splines (NURBS) to develop efficient and robust numerical techniques to deal with the problems of incompressibility in the fields of solid and fluid mechanics. Towards this, two types of formulations, mixed Galerkin and least-squares, are studied. During the first phase of this work, mixed Galerkin formulations, in the context of isogeometric analysis, are presented. Two-field and three-field mixed variational formulations - in both small and large strains - are presented to obtain accurate numerical solutions for the problems modelled with nearly incompressible and elasto-plastic materials. The equivalence of the two mixed formulations, for the considered material models, is derived; and the computational advantages of using two-field formulations are illustrated. Performance of these formulations is assessed by studying several benchmark examples. The ability of the mixed methods, to accurately compute limit loads for problems involving elastoplastic material models; and to deal with volumetric locking, shear locking and severe mesh distortions in finite strains, is illustrated. Later, finite element formulations are developed by combining least-squares and isogeometric analysis in order to extract the best of both. Least-squares finite element methods (LSFEMs) based on the use of governing differential equations directly - without the need to reduce them to equivalent lower-order systems - are developed for compressible and nearly incompressible elasticity in both the small and finite strain regimes; and incompressible Navier-Stokes. The merits of using Gauss-Newton scheme instead of Newton-Raphson method to solve the underlying nonlinear equations are presented. The performance of the proposed LSFEMs is demonstrated with several benchmark examples from the literature. Advantages of using higher-order NURBS in obtaining optimal convergence rates for non-norm-equivalent LSFEMs; and the robustness of LSFEMs, for Navier-Stokes, in obtaining accurate numerical solutions without the need to incorporate any artificial stabilisation techniques, are demonstrated

Least-squares mixed finite elements for geometrically nonlinear solid mechanics

The computation of reliable results using finite elements is a major engineering goal. Under the assumption of a linear elastic theory many stable and reliable (standard and mixed) finite elements have been developed. Unfortunately, in the geometrically nonlinear regime, e.g. applying these elements in the field of incompressible, hyperelastic materials, problems can occur. A possible approach to circumvent these issues might be the least-squares mixed finite element method. Therefore, in this thesis, a mixed least-squares formulation for hyperelastic materials in the field of solid mechanics is provided, investigated and valuated. To create a theoretical basis the continuum mechanical background is outlined, the necessary physical quantities are introduced and the construction of suitable interpolation functionsis shown. Furthermore, the general procedure for the construction of a least-squares functional is described and applied for hyperelastic material laws based on a free energy function. Basis for the proposed least-squares element formulation is a div-grad first-order system consisting of the equilibrium condition, the constitutive equation and a stress symmetry condition, all written in a residual form. The solution variables (displacements and stresses) are, dependent on the element type, interpolated using different approximation spaces. The performance of the provided elements is investigated and compared to standard and mixed Galerkin elements by extensive numerical studies with respect to e.g. bending dominated problems, incompressibility, stability issues, convergence of the field quantities and adaptivity. Furthermore, the crucial influence of weighting is discussed. Finally, the results are evaluated and the used elements are assessed.Ein Hauptziel im Bereich des Ingenieurwesens ist die Berechnung vertrauenswürdiger Ergebnisse mit Hilfe der Methode der finiten Elemente. Unter Annahme einer linear elastischen Theorie wurden hierzu bereits viele stabile und zuverlässige standard und gemischte finite Elemente entwickelt. Es hat sich jedoch herausgestellt, dass bei einigen dieser Elemente, unter anderem angewandt auf inkompressible, hyperelastische Materialien, Probleme auftreten. Ein möglicher Ansatz um diese Probleme zu umgehen ist möglicherweise die gemischte least-squares finite Elemente Methode. Daher wird in Rahmen dieser Arbeit eine gemischte least-squares Formulierung für hyperelastische Materialien vorgestellt, untersucht und bewertet. Um eine theoretische Basis zu schaffen wird zuerst ein kontinuumsmechanischer Rahmen geschaffen, die nötigen physikalischen Größen werden eingeführt und die Konstruktion geeigneter Interpolationsfunktionen wird gezeigt. Im Folgenden wird das allgemeine Vorgehen zur Konstruktion eines least-squares Funktionals beschrieben und angewandt auf hyperelastische Materialien in der Festkörpermechanik basierend auf freien Energiefunktionen. Die Basis für die least-squares Formulierung stellt ein div-grad System erster Ordnung dar, bestehend aus der Gleichgewichtsbedingung, einem Materialgesetz und einer zusätzlichen Bedingung für die Einhaltung einer Spannungssymmetrie. Die Gleichungen liegen hierbei in einer residualen Form vor. Die Lösungsvariablen sind, im Rahmen dieser Arbeit, die Verschiebungen und die Spannungen welche, abhängig vom Elementtyp, mit unterschiedlichen Interpolationsfunktionen interpoliert werden. Die Performanz der entwickelten Elemente wird im Folgenden mit extensiven numerischen Studien untersucht, welche sich unter anderem mit biegedominierten Problemen, Inkompressibilität, Untersuchung von Stabilitätspunkten und der allgemeinen Konvergenz der Lösungsvariablen beschäftigen. Zur Bewertung der Ergebnisse werden diese mit Lösungen verglichen, welche durch standard und gemischte Galerkin Elemente berechnet wurden. Darüber hinaus wird der starke Einfluss der Wichtungsfaktoren auf die Qualität der Lösungen diskutiert. Abschließend werden die Ergebnisse ausgewertet und die entwickelten Elemente bewertet