Crack face contact for a hexahedral-based XFEM formulation

Mueller-Hoeppe, Dana; Wriggers, Peter; Löhnert, Stefan

doi:10.1007/s00466-012-0701-2

Cited by 25 publications

(13 citation statements)

References 28 publications

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“…A variety of contact modeling methods have been explored in the XFEM. A penalty method has been employed to model small strain frictional contact problems using XFEM, for example, by . The XFEM framework has been used to investigate interface cohesion effects on the mechanical performance of nano‐structures .…”

Section: Introductionmentioning

confidence: 99%

Level set topology optimization of structural problems with interface cohesion

Behrou

Lawry

Maute

2017

Numerical Meth Engineering

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This paper presents a finite element topology optimization framework for the design of two-phase structural systems considering contact and cohesion phenomena along the interface. The geometry of the material interface is described by an explicit level set method, and the structural response is predicted by the extended finite element method. In this work, the interface condition is described by a bilinear cohesive zone model on the basis of the traction-separation constitutive relation. The non-penetration condition in the presence of compressive interface forces is enforced by a stabilized Lagrange multiplier method. The mechanical model assumes a linear elastic isotropic material, infinitesimal strain theory, and a quasi-static response. The optimization problem is solved by a nonlinear programming method, and the design sensitivities are computed by the adjoint method. The performance of the presented method is evaluated by 2D and 3D numerical examples. The results obtained from topology optimization reveal distinct design characteristics for the various interface phenomena considered. In addition, 3D examples demonstrate optimal geometries that cannot be fully captured by reduced dimensionality. The optimization framework presented is limited to two-phase structural systems where the material interface is coincident in the undeformed configuration, and to structural responses that remain valid considering small strain kinematics.LEVEL SET TOPOLOGY OPTIMIZATION WITH INTERFACE COHESION 991 known as topology optimization. To provide a high level of design freedom, a topology optimization framework is used in this paper.The interface conditions considered in this study are inherently nonlinear. For frictionless contact, interfacial forces act to prevent the penetration of bodies but vanish during separation. Material cohesion provides resistance to shear and normal separation of joined materials, but can result in rapid delamination when the cohesive limit is surpassed. Due to their complex behavior, problems with contact phenomena have only been considered in a few 2D topology optimization studies. This paper presents a novel topology optimization method for 2D and 3D problems involving interface cohesion and delamination.Density methods, such as the solid isotropic material with penalization approach, are the most common method of describing the geometry in topology optimization. The solid isotropic material with penalization approach was originally developed by [1,2] and describes the geometry of a body by defining the material distribution in the design domain as a function of design variables. A fictitious porous material with density, 0 ⩽ ⩽ 1, defines a continuous transition between two or more materials. For more information and an overview of recent developments, the reader is referred to [3][4][5]. The geometry of an interface is represented by either large spatial gradients or by jumps in the density fields, depending on the discretization of the density distribution and the method of enforcing converg...

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Section: Introductionmentioning

confidence: 99%

Level set topology optimization of structural problems with interface cohesion

Behrou

Lawry

Maute

2017

Numerical Meth Engineering

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“…Modeling contact problems with the XFEM has shown great promise considering both friction and sliding contact. Assuming infinitesimal strains, Liu and Borja (2008) and Mueller-Hoeppe et al (2012) enforce the contact conditions by a penalty method. The approaches of Geniaut et al (2007) and Amdouni et al (2012) are based on a Lagrange multiplier formulation.…”

Section: Introductionmentioning

confidence: 99%

Level set topology optimization of problems with sliding contact interfaces

Lawry

Maute

2015

Struct Multidisc Optim

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This paper introduces a topology optimization method for the design of two-component structures and two-phase material systems considering sliding contact and separation along interfaces. The geometry of the contact interface is described by an explicit level set method which allows for both shape and topological changes in the optimization process. The mechanical model assumes infinitesimal strains, a linear elastic material behavior, and a quasi-static response. The contact conditions are enforced by a stabilized Lagrange multiplier method and an active set strategy. The mechanical model is discretized by the extended finite element method which retains the crispness of the level set geometry description and allows for the convenient integration of the weak form of the contact conditions at the phase boundaries. The formulation of the optimization problem is regularized by introducing a perimeter penalty into the objective function. The optimization problem is solved by a nonlinear programming scheme computing the design sensitivities by the adjoint method. The main characteristics of the proposed method are studied by numerical examples in two dimensions. Consideration of contact leads to the formation of barb-type features that increase the interface stiffness. The numerical results further demonstrate the significant difference in the optimized geometries when assuming perfect bonding versus considering contact.

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“…Since the initial work, the penalty method, Nitsche's method, and the method of Lagrange multipliers have been pursued for solving the contact problems in unfitted frameworks. The penalty formulation for the contact between the open crack faces was utilized in a few works [KN06,LB08,MHWL12]. Nitsche's method has also been proposed for solving the frictional contact problem in the unfitted framework [CSS11].…”

Section: Introductionmentioning

confidence: 99%

A Generalized Multigrid Method for Solving Contact Problems in Lagrange Multiplier based Unfitted Finite Element Method

Kothari,

Krause

2021

Preprint

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Internal interfaces in a domain could exist as a material defect or they can appear due to propagations of cracks. Discretization of such geometries and solution of the contact problem on the internal interfaces can be computationally challenging. We employ an unfitted Finite Element (FE) framework for the discretization of the domains and develop a tailored, globally convergent, and efficient multigrid method for solving contact problems on the internal interfaces. In the unfitted FE methods, structured background meshes are used and only the underlying finite element space has to be modified to incorporate the discontinuities. The non-penetration conditions on the embedded interfaces of the domains are discretized using the method of Lagrange multipliers. We reformulate the arising variational inequality problem as a quadratic minimization problem with linear inequality constraints. Our multigrid method can solve such problems by employing a tailored multilevel hierarchy of the FE spaces and a novel approach for tackling the discretized non-penetration conditions. We employ pseudo-L 2 projection-based transfer operators to construct a hierarchy of nested FE spaces from the hierarchy of non-nested meshes. The essential component of our multigrid method is a technique that decouples the linear constraints using an orthogonal transformation of the basis. The decoupled constraints are handled by a modified variant of the projected Gauss-Seidel method, which we employ as a smoother in the multigrid method. These components of the multigrid method allow us to enforce linear constraints locally and ensure the global convergence of our method. We will demonstrate the robustness, efficiency, and level independent convergence property of the proposed method for Signorini's problem and two-body contact problems.

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Crack face contact for a hexahedral-based XFEM formulation

Cited by 25 publications

References 28 publications

Level set topology optimization of structural problems with interface cohesion

Level set topology optimization of structural problems with interface cohesion

Level set topology optimization of problems with sliding contact interfaces

A Generalized Multigrid Method for Solving Contact Problems in Lagrange Multiplier based Unfitted Finite Element Method

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