Progress in mixed Eulerian-Lagrangian finite element simulation of forming processes

A review is given of a mixed Eulerian-Lagrangian finite element method for simulation of forming processes. This method permits incremental adaptation of nodal point locations independently from the actual material displacements. Hence numerical difficulties due to large element distortions, as may occur when the updated Lagrange method is applied, can be avoided. Movement of (free) surfaces can be taken into account by adapting nodal surface points in a way that they remain on the surface. Hardening and other deformation path dependent properties are determined by incremental treatment of convective terms. A local and a weighed global smoothing procedure is introduced in order to avoid numerical instabilities and numerical diffusion. Prediction of contact phenomena such as gap openning and/or closing and sliding with friction is accomplished by a special contact element. The method is demonstrated by simulations of an upsetting process and a wire drawing process

A priori error for unilateral contact problems with Lagrange multiplier and IsoGeometric Analysis

In this paper, we consider unilateral contact problem without friction between a rigid body and deformable one in the framework of isogeometric analysis. We present the theoretical analysis of the mixed problem using an active-set strategy and for a primal space of NURBS of degree $p$ and $p-2$ for a dual space of B-Spline. A inf-sup stability is proved to ensure a good property of the method. An optimal a priori error estimate is demonstrated without assumption on the unknown contact set. Several numerical examples in two- and three-dimensional and in small and large deformation demonstrate the accuracy of the proposed method

A finite element framework for modeling internal frictional contact in three-dimensional fractured media using unstructured tetrahedral meshes

AbstractThis paper introduces a three-dimensional finite element (FE) formulation to accurately model the linear elastic deformation of fractured media under compressive loading. The presented method applies the classic Augmented Lagrangian(AL)-Uzawa method, to evaluate the growth of multiple interacting and intersecting discrete fractures. The volume and surfaces are discretized by unstructured quadratic triangle-tetrahedral meshes; quarter-point triangles and tetrahedra are placed around fracture tips. Frictional contact between crack faces for high contact precisions is modeled using isoparametric integration point-to-integration point contact discretization, and a gap-based augmentation procedure. Contact forces are updated by interpolating tractions over elements that are adjacent to fracture tips, and have boundaries that are excluded from the contact region. Stress intensity factors are computed numerically using the methods of displacement correlation and disk-shaped domain integral. A novel square-root singular variation of the penalty parameter near the crack front is proposed to accurately model the contact tractions near the crack front. Tractions and compressive stress intensity factors are validated against analytical solutions. Numerical examples of cubes containing one, two, twenty four and seventy interacting and intersecting fractures are presented

Stabilized four-node tetrahedron with nonlocal pressure for modeling hyperelastic materials

Non-linear hyperelastic response of reinforced elastomers is modeled using a novel three-dimensional mixed finite element method with a nonlocal pressure field. The element is unconditionally convergent and free of spurious pressure modes. Nonlocal pressure is obtained by an implicit gradient technique and obeys the Helmholtz equation. Physical motivation for this nonlocality is shown. An implicit finite element scheme with consistent linearization is presented. Finally, several hyperelastic examples are solved to demonstrate the computational algorithm including the inf–sup and verifications test

Evaluation of Accuracy and Efficiency of Numerical Methods for Contact Problems

Tato diplomová práce se zabývá variačními metodami, které umožňují formulovat problém kontaktu lineárně pružného tělesa bez tření jako nepodmíněnou variační rovnost, která může být posléze diskretizována a řešena metodou konečných prvků. Hlavní důraz je kladen na Nitscheho metody podle Wriggerse a Zavariseho [56] a podle Fabrého, Pousina a Renarda [15]. V současné době nejrozšířenější konečněprvkové softwarové balíky, jako jsou ANSYS, ABAQUS a COMSOL, využívají pro modelování kontaktu především standardní metody penalty a smíšené metody [57, Kapitola 1.1.1, p.7]. Ukazuje se, že právě Nitscheho metody mají potenciál překonat klasické obtíže spojené se standardními metodami penalty a smíšenými metodami. Na rozdíl od metod penalty jsou Nitscheho metody konzistentní a kontaktní okrajové podmínky jsou vynuceny přesně (na teoretické úrovni). Je také možné využít mnohem menší hodnotu parametru penalty, čímž se lze vyhnout problémům spojeným se špatným podmíněním úlohy, charakteristickým pro metody penalty. Nitscheho metoda ale současně nevyžaduje přidání žádných dalších neznámých (Lagrangeových multiplikátorů) a výsledný diskrétní systém tak není nadbytečně rozšířen, jako je tomu v případě smíšených metod. Oproti smíšeným metodám také není třeba věnovat pozornost splnění Babuškovy-Brezziho podmínky. V této diplomové práci se ukazuje, že analyzované Nitscheho metody úzce souvisejí s metodami penalty a metodou augmentovaného lagrangiánu. V práci jsou prezentovány slabé formulace těchto metod a zkoumají se rozdíly mezi formulací Nitscheho metody podle Wriggerse a podle Fabrého, Pousina a Renarda. Všechny metody jsou implementovány do prostředí FEniCS (výpočetní platforma pro řešení parciálních diferenciálních rovnic metodou konečných prvků) a jejich přesnost a výkonnost se testuje na různých dvourozměrných a trojrozměrných problémech kontaktu lineárně pružného tělesa s dokonale tuhou rovinou. Na jednoduchém dvourozměrném příkladu je ukázáno, že funkce, kterou získáme jako levou stranu diskretizované slabé formy Wriggersovy varianty Nitscheho metody, není spojitá vzhledem k neznámým stupňům volnosti. Tento poznatek vysvětluje problémy s konvergencí Newtonovy metody při řešení Wriggersovou variantou Nitscheho metodou, které jsme zaznamenali při numerických experimentech.This thesis is concerned with various methods that allow us to formulate the frictionless linear elastic contact problems as an unconstrained variational equality, which is then discretised and solved with the finite element method. The main focus is on Nitsche methods in the forms used respectively by Wriggers and Zavarise [56] and Fabré, Pousin and Renard [15]. Currently, standard penalty and mixed methods are dominant in the modern leading finite element software packages such as ANSYS, ABAQUS and COMSOL [57, Chapter 1.1.1, p.7]. Nitsche methods display a potential to overcome classic drawbacks of the penalty and mixed methods. Unlike penalty methods, Nitsche methods are consistent, and contact boundary conditions are enforced precisely (on the theoretical level). Also, a significantly smaller value of the penalty parameter is necessary and the possible ill-conditioning, so characteristic for penalty methods, is thus avoided. At the same time, no additional unknowns (Lagrange multipliers) are introduced; thus, the corresponding discrete system is not enlarged, and one does not have to worry about the Babuška-Brezzi condition. In this thesis was shown that the analysed Nitsche methods are closely related to penalty methods and the augmented Lagrangian method. The weak forms of all these methods are presented, and differences between Wriggers' version and Fabré, Pousin and Renard's version of Nitsche method are investigated. All methods are implemented in FEniCS (the computational platform for solving partial differential equations with the finite element method), and their accuracy and efficiency is tested on various two- and three-dimensional numerical examples of contact of an elastic body with a rigid plane. By means of the simple two-dimensional example it is shown that the function obtained as the left-hand side of the discretised weak form of the Nitsche-Wriggers method is not continuous with respect to the unknown displacement DOFs. This finding explains the convergence problems (of Newton's method) that the Nitsche-Wriggers method suffers from, unlike other investigated methods

A Mixed Eulerian-Lagrangian Model for the Analysis of Dynamic Fracture

National Science Foundation Grant MEA 84-0065