This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

A variational free-discontinuity formulation of brittle fracture was given by Francfortand Marigo [1], where the total energy is minimized with respect to the crackgeometry and the displacement field simultaneously. The entire evolution of cracksincluding their initiation and branching is determined by this minimization principlerequiring no further criterion. However, a direct numerical discretization of themodel faces considerable difficulties as the displacement field is discontinuous inthe presence of cracks.

A Continuum Phase Field Model for Fracture

THE best adjective to describe this work is \"sweep11 ing.\" The range of subject matter is so broad that it can almost be described as containing everything except fuzzy set theory. Included are explicit discussions of the basics of probability (relegated to an appendix); the practice of problem solving; testing hypotheses about statistical parameters; calculating and interpreting confidence limits; tolerance limits and prediction limits; setting up and interpreting control charts; design of experiments; analysis of variance; line and surface fitting; and maximum likelihood procedures. If you can think of something that is not in this list, then it probably means I have overlooked it.

Experiments: Planning, Analysis, and Parameter Design Optimization

Damage and fracture algorithm using the screened Poisson equation and local remeshing

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Methods for the prediction of fatigue delamination growth in composites and adhesive bonds: A critical review

The numerical simulation of fatigue crack growth using extended finite element method

Numerical simulation of functionally graded cracked plates using NURBS based XIGA under different loads and boundary conditions

An adaptive multiscale phase field method for brittle fracture

An investigation of fatigue crack growth of interfacial cracks in bi-layered materials using the extended finite element method is presented. The bi-material consists of two layers of dissimilar materials. The bottom layer is made of aluminium alloy while the upper one is made of functionally graded material (FGM). The FGM layer consists of 100 % aluminium alloy on the left side and 100 % ceramic (alumina) on the right side. The gradation in material property of the FGM layer is assumed to be exponential from the alloy side to the ceramic side. The domain based interaction integral approach is extended to obtain the stress intensity factors for an interfacial crack under thermo-mechanical load. The edge and centre cracks are taken at the interface of bi-layered material. The fatigue life of the interface crack plate is obtained using the Paris law of fatigue crack growth under cyclic mode-I, mixed-mode and thermal loads. This study reveals that the crack propagates into the FGM layer under all types of loads.

B.K. Mishra

Papers

The numerical simulation of fatigue crack growth using extended finite element method

Numerical simulation of functionally graded cracked plates using NURBS based XIGA under different loads and boundary conditions

An adaptive multiscale phase field method for brittle fracture

Fatigue crack growth simulations of interfacial cracks in bi-layered FGMs using XFEM

Stochastic fatigue crack growth simulation of interfacial crack in bi-layered FGMs using XIGA