Fundamentals of Enriched Finite Element Methods

Front Cover
Fundamentals of Enriched Finite Element Methods
Copyright
Contents
Preface
1 Introduction
	1.1 Enriched finite element methods
	1.2 Origins and milestones of e-FEMs
	References
I Fundamentals
	2 The finite element method
		2.1 Linear elastostatics in 1-D
			2.1.1 The strong form
			2.1.2 The weak (or variational) form
				2.1.2.1 Sobolev spaces
				2.1.2.2 Non-homogeneous Dirichlet boundary conditions
			2.1.3 The Galerkin formulation
				2.1.3.1 Orthogonality of Galerkin error
			2.1.4 The finite element discrete equations
			2.1.5 The isoparametric mapping
			2.1.6 A priori error estimates
			2.1.7 A posteriori error estimate
		2.2 The elastostatics problem in higher dimensions
			2.2.1 Strong form
			2.2.2 Weak form
			2.2.3 Principle of virtual work
			2.2.4 Discrete formulation
			2.2.5 Voigt notation
			2.2.6 Isoparametric formulation in higher dimensions
		2.3 Heat conduction
		2.4 Problems
		References
	3 The p-version of the finite element method
		3.1 p-FEM in 1-D
			3.1.1 A priori error estimates
		3.2 p-FEM in 2-D
			3.2.1 Basis functions for quadrangles
			3.2.2 Basis functions for triangles
		3.3 Non-homogeneous essential boundary conditions
			3.3.1 Interpolation at Gauss–Lobatto quadrature points
			3.3.2 Projection on the space of edge functions
		3.4 Problems
		References
	4 The Generalized Finite Element Method
		4.1 Finite element approximations
		4.2 Generalized FEM approximations in 1-D
			4.2.1 Selection of enrichment functions
			4.2.2 What makes the GFEM work
		4.3 Applications of the GFEM
		4.4 Shifted and scaled enrichments
		4.5 The p-version of the GFEM
			4.5.1 High-order GFEM approximations for a strong discontinuity
		4.6 GFEM approximation spaces
		4.7 Exercises
		References
	5 Discontinuity-enriched finite element formulations
		5.1 A weak discontinuity in 1-D
		5.2 A strong discontinuity in 1-D
		5.3 Relationship to GFEM
		5.4 The discontinuity-enriched FEM in multiple dimensions
			5.4.1 Treatment of nonzero essential boundary conditions
			5.4.2 Hierarchical space
		5.5 Convergence
		5.6 Weak and strong discontinuities
		5.7 Recovery of field gradients
		References
II Applications
	6 GFEM approximations for fractures
		6.1 Governing equations: 3-D elasticity
			6.1.1 Weak form
		6.2 GFEM approximation for fractures
			6.2.1 Approximation of ̂u
				6.2.1.1 High-order approximations
			6.2.2 Approximation of ̃̃u
			6.2.3 Cohesive fracture problems
				6.2.3.1 High-order approximations
			6.2.4 Approximation of ̆u
				6.2.4.1 Elasticity solution in the neighborhood of a crack front
				6.2.4.2 Oden and Duarte branch enrichment functions
				6.2.4.3 Belytschko and Black branch enrichment functions
			6.2.5 Topological and geometrical singular enrichment
			6.2.6 Discrete equilibrium equations
		6.3 Convergence of linear GFEM approximations: 2-D edge crack
			6.3.1 Topological enrichment
			6.3.2 Comparison with best-practice FEM
			6.3.3 Geometrical enrichment
		6.4 Convergence of linear GFEM approximations: 3-D edge crack
		References
	7 Generalized enrichment functions for weak discontinuities
		7.1 Formulation
			7.1.1 Linear elastostatics
			7.1.2 Heat conduction
			7.1.3 Discrete equations
			7.1.4 Enrichment functions for weak discontinuities
				7.1.4.1 The distance function
				7.1.4.2 The distance function with smoothing
				7.1.4.3 The ridge function
				7.1.4.4 Corrected enrichments
			7.1.5 Enrichment performance
		7.2 Discussion and further reading
		References
	8 Immerse boundary (fictitious domain) problems
		8.1 Formulation
			8.1.1 Treatment of boundary conditions
			8.1.2 An immersed popcorn example
		References
	9 Non-conforming mesh coupling and contact
		9.1 Formulation
		9.2 Examples
			9.2.1 Infinite plate with a circular hole
			9.2.2 Hertzian contact
		9.3 Further reading
		References
	10 Interface-enriched topology optimization
		10.1 Formulation
			10.1.1 Enriched finite element analysis
			10.1.2 Design space
			10.1.3 Optimization
		10.2 Compliance minimization
			3-D cantilever beam example
		10.3 Fracture resistance
			Biaxial tension example
			L-shaped bracket example
		10.4 Discussion and further reading
		References
III Computational aspects
	11 Stability of approximations
		11.1 Conditioning control of GFEM matrices
			11.1.1 Scaling of global matrices
			11.1.2 Conditioning control of Heaviside enrichments
				11.1.2.1 Conditioning of GFEM approximation for a strong discontinuity
				11.1.2.2 Extension to higher dimensions
			11.1.3 Conditioning control of singular enrichments
				11.1.3.1 Discontinuous shifting of enrichment functions
			11.1.4 Well-conditioned first-order GFEM approximations
			11.1.5 Solution of singular systems of equations
			11.1.6 Two-dimensional inclined edge crack
		11.2 Stability of interface-enriched generalized finite element formulations
			11.2.1 IGFEM scaling in 1-D
			11.2.2 IGFEM stability in higher dimensions
		References
	12 Computational aspects
		12.1 A basic FEM code structure
		12.2 e-FEM considerations
			12.2.1 Computational geometry
				12.2.1.1 Detecting discontinuities
				12.2.1.2 Element partitioning
				12.2.1.3 Discontinuity representation
				12.2.1.4 Propagating discontinuities
			12.2.2 Numerical integration
		References
	13 Approximation theory for partition of unity methods
		13.1 Approximation theory for the GFEM
		13.2 A priori error estimates for partition of unity approximations
		References
A Recollections of the origins of the GFEM
	A.1 The hp-cloud method
	A.2 The name partition of unity method is coined
	A.3 A partition of unity method for problems with singularities
	A.4 Wrapping up my PhD at UT Austin
	A.5 Work on partition of unity methods at COMCO and Altair Engineering
		A.5.1 The Element Partition Method (EPM)
		A.5.2 Collaboration with Ivo Babuška and Tinsley Oden
	A.6 Early work on partition of unity methods at Texas A&M University
	A.7 Early work on partition of unity methods at Northwestern University
	References
Index
Back Cover