Showing papers on "Extended finite element method published in 2015"

CutFEM: Discretizing geometry and partial differential equations

Abstract: We discuss recent advances on robust unfitted finite element methods on cut meshes. These methods are designed to facilitate computations on complex geometries obtained, for example, from computer- ...

Least-Squares Finite Element Methods

Abstract: Least-squares finite element methods are an attractive class of methods for the numerical solution of partial differential equations. They are motivated by the desire to recover, in general settings, the advantageous features of Rayleigh�Ritz methods such as the avoidance of discrete compatibility conditions and the production of symmetric and positive definite discrete systems. The methods are based on the minimization of convex functionals that are constructed from equation residuals. This paper focuses on theoretical and practical aspects of least-square finite element methods and includes discussions of what issues enter into their construction, analysis, and performance. It also includes a discussion of some open problems.

Introduction to nonlinear finite element analysis

Abstract: Preliminary concepts.- Nonlinear Finite Element Analysis Procedure.- Finite Element Analysis for Nonlinear Elastic Systems.- Finite Element Analysis for Elastoplastic Problems.- Finite Element Analysis for Contact Problems.

The Finite Cell Method: A Review in the Context of Higher-Order Structural Analysis of CAD and Image-Based Geometric Models

Abstract: The finite cell method is an embedded domain method, which combines the fictitious domain approach with higher-order finite elements, adaptive integration, and weak enforcement of unfitted essential boundary conditions. Its core idea is to use a simple unfitted structured mesh of higher-order basis functions for the approximation of the solution fields, while the geometry is captured by means of adaptive quadrature points. This eliminates the need for boundary conforming meshes that require time-consuming and error-prone mesh generation procedures, and opens the door for a seamless integration of very complex geometric models into finite element analysis. At the same time, the finite cell method achieves full accuracy, i.e. optimal rates of convergence, when the mesh is refined, and exponential rates of convergence, when the polynomial degree is increased. Due to the flexibility of the quadrature based geometry approximation, the finite cell method can operate with almost any geometric model, ranging from boundary representations in computer aided geometric design to voxel representations obtained from medical imaging technologies. In this review article, we first provide a concise introduction to the basics of the finite cell method. We then summarize recent developments of the technology, with particular emphasis on the research topics in which we have been actively involved. These include the finite cell method with B-spline and NURBS basis functions, the treatment of geometric nonlinearities for large deformation analysis, the weak enforcement of boundary and coupling conditions, and local refinement schemes. We illustrate the capabilities and advantages of the finite cell method with several challenging examples, e.g. the image-based analysis of foam-like structures, the patient-specific analysis of a human femur bone, the analysis of volumetric structures based on CAD boundary representations, and the isogeometric treatment of trimmed NURBS surfaces. We conclude our review by briefly discussing some key aspects for the efficient implementation of the finite cell method.

Extended Finite Element Method: Theory and Applications

Abstract: Introduces the theory and applications of the extended finite element method (XFEM) in the linear and nonlinear problems of continua, structures and geomechanics • Explores the concept of partition of unity, various enrichment functions, and fundamentals of XFEM formulation. • Covers numerous applications of XFEM including fracture mechanics, large deformation, plasticity, multiphase flow, hydraulic fracturing and contact problems • Accompanied by a website hosting source code and examples

An Introduction to Nonlinear Finite Element Analysis: with applications to heat transfer, fluid mechanics, and solid mechanics

Abstract: 1. General Introduction and Mathematical Preliminaries 2. Elements of Nonlinear Continuum Mechanics 3. The Finite Element Method: A Review 4. One-Dimensional Problems Involving a Single Variable 5. Nonlinear Bending of Straight Beams 6. Two-Dimensional Problems Involving a Single Variable 7. Nonlinear Bending of Elastic Plates 8. Nonlinear Bending of Elastic Shells 9. Finite Element Formulations of Solid Continua 10. Weak-Form Finite Element Models of Flows of Viscous Incompressible Fluids 11. Least-Squares Finite Element Models of Flows of Viscous Incompressible Fluids Appendix 1: Solution Procedures for Linear Equations Appendix 2: Solution Procedures for Nonlinear Equations

A Weak Galerkin Finite Element Method for the Maxwell Equations

Abstract: This paper introduces a numerical scheme for the time-harmonic Maxwell equations by using weak Galerkin (WG) finite element methods. The WG finite element method is based on two operators: discrete weak curl and discrete weak gradient, with appropriately defined stabilizations that enforce a weak continuity of the approximating functions. This WG method is highly flexible by allowing the use of discontinuous approximating functions on arbitrary shape of polyhedra and, at the same time, is parameter free. Optimal-order of convergence is established for the WG approximations in various discrete norms which are either $$H^1$$H1-like or $$L^2$$L2 and $$L^2$$L2-like. An effective implementation of the WG method is developed through variable reduction by following a Schur-complement approach, yielding a system of linear equations involving unknowns associated with element boundaries only. Numerical results are presented to confirm the theory of convergence.

Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations

Abstract: We present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem. Relying on the equilibration by the mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree. Maximal local overestimation is guaranteed as well. Numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous Galerkin scheme.

Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction–diffusion problem

Abstract: In this article, a finite difference/finite element algorithm, which is based on a finite difference approximation in time direction and finite element method in spatial direction, is presented and discussed to cast about for the numerical solutions of a time-fractional fourth-order reaction–diffusion problem with a nonlinear reaction term. To avoid the use of higher-order elements, the original problem with spatial fourth-order derivative need to be changed into a second-order coupled system by introducing an intermediate variable σ = Δ u . Then the fully discrete finite element scheme is formulated by using a finite difference approximation for time fractional and integer derivatives and finite element method in spatial direction. The unconditionally stable result in the norm, which just depends on initial value and source item, is derived. Some a priori estimates of L 2 -norm with optimal order of convergence O ( Δ t 2 − α + h m + 1 ) , where Δ t and h are time step length and space mesh parameter, respectively, are obtained. To confirm the theoretical analysis, some numerical results are provided by our method.

Space-Time Finite Element Methods for Parabolic Problems

Abstract: Abstract We propose and analyze a space-time finite element method for the numerical solution of parabolic evolution equations. This approach allows the use of general and unstructured space-time finite elements which do not require any tensor product structure. The stability of the numerical scheme is based on a stability condition which holds for standard finite element spaces. We also provide related a priori error estimates which are confirmed by numerical experiments.

Extended finite element method in computational fracture mechanics: a retrospective examination

Abstract: In this paper, we provide a retrospective examination of the developments and applications of the extended finite element method (X-FEM) in computational fracture mechanics. Our main attention is placed on the modeling of cracks (strong discontinuities) for quasistatic crack growth simulations in isotropic linear elastic continua. We provide a historical perspective on the development of the method, and highlight the most important advances and best practices as they relate to the formulation and numerical implementation of the X-FEM for fracture problems. Existing challenges in the modeling and simulation of dynamic fracture, damage phenomena, and capturing the transition from continuum-to-discontinuum are also discussed.

A nonlinear, transient finite element method for coupled solvent diffusion and large deformation of hydrogels

Abstract: Hydrogels are capable of coupled mass transport and large deformation in response to external stimuli. In this paper, a nonlinear, transient finite element formulation is presented for initial boundary value problems associated with swelling and deformation of hydrogels, based on a nonlinear continuum theory that is consistent with classical theory of linear poroelasticity. A mixed finite element method is implemented with implicit time integration. The incompressible or nearly incompressible behavior at the initial stage imposes a constraint to the finite element discretization in order to satisfy the Ladyzhenskaya–Babuska–Brezzi (LBB) condition for stability of the mixed method, similar to linear poroelasticity as well as incompressible elasticity and Stokes flow; failure to choose an appropriate discretization would result in locking and numerical oscillations in transient analysis. To demonstrate the numerical method, two problems of practical interests are considered: constrained swelling and flat-punch indentation of hydrogel layers. Constrained swelling may lead to instantaneous surface instability for a soft hydrogel in a good solvent, which can be regulated by assuming a stiff surface layer. Indentation relaxation of hydrogels is simulated beyond the linear regime under plane strain conditions, in comparison with two elastic limits for the instantaneous and equilibrium states. The effects of Poisson’s ratio and loading rate are discussed. It is concluded that the present finite element method is robust and can be extended to study other transient phenomena in hydrogels.

Real-time control of soft-robots using asynchronous finite element modeling

Abstract: Finite Element analysis can provide accurate deformable models for soft-robots. However, using such models is very difficult in a real-time system of control. In this paper, we introduce a generic solution that enables a high-rate control and that is compatible with strong real-time constraints. From a Finite Element analysis, computed at low rate, an inverse model of the robot outputs the setpoint values for the actuator in order to obtain a desired trajectory. This inverse problem uses a QP (quadratic-programming) algorithm based on the equations set by the Finite Element Method. To improve the update rate performances, we propose an asynchronous simulation framework that provides a better trade-off between the deformation accuracy and the computational burden. Complex computations such as accurate FEM deformations are done at low frequency while the control is performed at high frequency with strong real-time constraints. The two simulation loops (high frequency and low frequency loops) are mechanically coupled in order to guarantee mechanical accuracy of the system over time. Finally, the validity of the multi-rate simulation is discussed based on measurements of the evolution in the QP matrix and an experimental validation is conducted to validate the correctness of the high-rate inverse model on a real robot.

Progressive delamination analysis of composite materials using XFEM and a discrete damage zone model

Abstract: The modeling of progressive delamination by means of a discrete damage zone model within the extended finite element method is investigated. This framework allows for both bulk and interface damages to be conveniently traced, regardless of the underlying mesh alignment. For discrete interfaces, a new mixed-mode force---separation relation, which accounts for the coupled interaction between opening and sliding modes, is proposed. The model is based on the concept of Continuum Damage Mechanics and is shown to be thermodynamically consistent. An integral-type nonlocal damage is adopted in the bulk to regularize the softening material response. The resulting nonlinear equations are solved using a Newton scheme with a dissipation-based arc-length constraint, for which an analytical Jacobian is derived. Several benchmark delamination studies, as well as failure analyses of a fiber/epoxy unit cell, are presented and discussed in detail. The proposed model is validated against available analytical/experimental data and is found to be robust and mesh insensitive.

Adhesive Selection for Single Lap Bonded Joints: Experimentation and Advanced Techniques for Strength Prediction

Abstract: The integrity of multi-component structures is usually determined by their unions. Adhesive-bonding is often used over traditional methods because of the reduction of stress concentrations, reduced weight penalty, and easy manufacturing. Commercial adhesives range from strong and brittle (e.g., Araldite® AV138) to less strong and ductile (e.g., Araldite® 2015). A new family of polyurethane adhesives combines high strength and ductility (e.g., Sikaforce® 7888). In this work, the performance of the three above-mentioned adhesives was tested in single lap joints with varying values of overlap length (LO). The experimental work carried out is accompanied by a detailed numerical analysis by finite elements, either based on cohesive zone models (CZM) or the extended finite element method (XFEM). This procedure enabled detailing the performance of these predictive techniques applied to bonded joints. Moreover, it was possible to evaluate which family of adhesives is more suited for each joint geometry. CZM revea...

Finite element multigrid method for multi-term time fractional advection diffusion equations

Abstract: In this paper, a class of multi-term time fractional advection diffusion equations (MTFADEs) is considered. By finite difference method in temporal direction and finite element method in spatial direction, two fully discrete schemes of MTFADEs with different definitions on multi-term time fractional derivative are obtained. The stability and convergence of these numerical schemes are discussed. Next, a V-cycle multigrid method is proposed to solve the resulting linear systems. The convergence of the multigrid method is investigated. Finally, some numerical examples are given for verification of our theoretical analysis.