Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume 1 One and Two Dimensional Elliptic and Maxwell Problems

by Demkowicz; Leszek

• ISBN13: 9781584886716
• ISBN10: 1584886714
• Edition: 1st
• Format: Hardcover
• Copyright: 2006-10-25
• Publisher: Chapman & Hall/

Summary

Offering the only existing finite element (FE) codes for Maxwell equations that support hp refinements on irregular meshes, Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume 1. One- and Two-Dimensional Elliptic and Maxwell Problems presents 1D and 2D codes and automatic hp adaptivity. This self-contained source discusses the theory and implementation of hp-adaptive FE methods, focusing on projection-based interpolation and the corresponding hp-adaptive strategy.The book is split into three parts, progressing from simple to more advanced problems. Part I examines the hp elements for the standard 1D model elliptic problem. The author develops the variational formulation and explains the construction of FE basis functions. The book then introduces the 1D code (1Dhp) and automatic hp adaptivity. This first part ends with a study of a 1D wave propagation problem. In Part II, the book proceeds to 2D elliptic problems, discussing two model problems that are slightly beyond standard-level examples: 3D axisymmetric antenna problem for Maxwell equations (example of a complex-valued, indefinite problem) and 2D elasticity (example of an elliptic system). The author concludes with a presentation on infinite elements - one of the possible tools to solve exterior boundary-value problems. Part III focuses on 2D time-harmonic Maxwell equations. The book explains the construction of the hp edge elements and the fundamental de Rham diagram for the whole family of hp discretizations.Next, it explores the differences between the elliptic and Maxwell versions of the 2D code, including automatic hp adaptivity. Finally, the book presents 2D exterior (radiation and scattering) problems and sample solutions using coupled hp finite/infinite elements.In Computing with hp-ADAPTIVE FINITE ELEMENTS, the information provided, including many unpublished details, aids in solving elliptic and Maxwell problems.

Author Biography

Leszek F. Demkowicz is associate director of the Institute for Computational Engineering and Sciences (ICES) and a professor in the department of aerospace engineering and engineering mechanics at the University of Texas at Austin, USA

Table of Contents

Part I 1D Problems	1	(132)
1 1D Model Elliptic Problem	3	(30)
1.1 A Two-Point Boundary Value Problem	3	(4)
1.1.1 Classical Formulation	3	(1)
1.1.2 Interface Problem	4	(1)
1.1.3 Weak (Variational) Formulation of the Problem	5	(2)
1.2 Algebraic Structure of the Variational Formulation	7	(2)
1.3 Equivalence with a Minimization Problem	9	(3)
1.4 Sobolev Space H¹(0, 1)	12	(3)
1.4.1 Distributional Derivatives	12	(2)
1.4.2 Finite Energy Solutions	14	(1)
1.4.3 Sobolev Space	14	(1)
1.5 Well Posedness of the Variational BVP	15	(4)
1.6 Examples from Mechanics and Physics	19	(8)
1.6.1 Elastic Bar	19	(3)
1.6.2 Heat Conduction	22	(2)
1.6.3 Vibrations of an Elastic Bar	24	(2)
1.6.4 Complex-Valued Problems	26	(1)
1.7 The Case with "Pure Neumann" BCs	27	(2)
Exercises	29	(4)
2 Galerkin Method	33	(8)
2.1 Finite-Dimensional Approximation of the VBVP	33	(3)
2.1.1 Petrov—Galerkin Method	35	(1)
2.1.2 Internal Approximation	35	(1)
2.1.3 Orthogonality Relation	36	(1)
2.1.4 The Ritz Method	36	(1)
2.2 Elementary Convergence Analysis	36	(3)
2.2.1 SPD Case	37	(1)
2.2.2 A General Positive-Definite Case	38	(1)
2.3 Comments	39	(1)
Exercises	39	(2)
3 1D hp Finite Element Method	41	(16)
3.1 1D hp Discretization	41	(5)
3.1.1 1D Master Element of Order p	42	(1)
3.1.2 Legendre Polynomials	43	(1)
3.1.3 1D Parametric Element of Arbitrary Order	44	(1)
3.1.4 1D hp Finite Element Space	45	(1)
3.2 Assembling Element Matrices into Global Matrices	46	(3)
3.3 Computing the Element Matrices	49	(2)
3.4 Accounting for the Dirichlet BC	51	(1)
3.5 Summary	52	(1)
3.6 Assignment 1: A Dry Run	53	(1)
Exercises	54	(3)
4 1D hp Code	57	(14)
4.1 Setting Up the 1D hp Code	57	(2)
4.1.1 Makefile	58	(1)
4.1.2 Data Types	58	(1)
4.1.3 The Coding Protocol	58	(1)
4.2 Fundamentals	59	(1)
4.3 Graphics	60	(5)
4.3.1 Postscript Echo	64	(1)
4.4 Element Routine	65	(1)
4.5 Assignment 2: Writing Your Own Processor	66	(3)
4.5.1 Natural Order of Nodes and Nodal d.o.f	67	(2)
Exercises	69	(2)
5 Mesh Refinements in 1D	71	(24)
5.1 The h-Extension Operator. Constrained Approximation Coefficients	71	(2)
5.2 Projection-Based Interpolation in 1D	73	(3)
5.2.1 p Refinements	73	(1)
5.2.2 p Unrefinements	74	(1)
5.2.3 Implementing Projection-Based Interpolation	75	(1)
5.2.4 Initiating d.o.f. for p and h Unrefinements	76	(1)
5.3 Supporting Mesh Refinements	76	(1)
5.4 Data-Structure-Supporting Routines	77	(2)
5.4.1 Natural Order of Elements	78	(1)
5.4.2 Reconstructing Element Nodal Connectivities	78	(1)
5.5 Programming Bells and Whistles	79	(3)
5.5.1 Interfacing with the Frontal Solver	80	(1)
5.5.2 Adaptive Integration	81	(1)
5.5.3 Choice of Shape Functions	82	(1)
5.6 Interpolation Error Estimates	82	(5)
5.6.1 p-Interpolation Error Estimate	83	(2)
5.6.2 hp-Interpolation Error Estimate	85	(2)
5.7 Convergence	87	(3)
5.7.1 Uniform h Refinements	87	(1)
5.7.2 Uniform p Refinements	88	(1)
5.7.3 Adaptive h Refinements	88	(1)
5.7.4 Adaptive p Refinements	89	(1)
5.7.5 Adaptive hp Refinements	89	(1)
5.7.6 Adaptive p Refinements with an Optimal Initial Mesh	89	(1)
5.8 Assignment 3: Studying Convergence	90	(1)
5.8.1 A Coercive Elliptic Problem	90	(1)
5.8.2 A Convection-Dominated Diffusion Problem	90	(1)
5.9 Definition of a Finite Element	91	(1)
Exercises	92	(3)
6 Automatic hp Adaptivity in 1D	95	(22)
6.1 The hp Algorithm	95	(9)
6.1.1 Projection-Based Interpolation in a Generalized Sense	96	(1)
6.1.2 The hp Algorithm	96	(4)
6.1.3 Unwanted h Refinements	100	(1)
6.1.4 Error Estimation. Stopping Criterion	101	(1)
6.1.5 Enforcing Optimal Refinements	102	(2)
6.2 Supporting the Optimal Mesh Selection	104	(3)
6.2.1 Determining the Optimal Refinement for an Element	105	(2)
6.3 Exponential Convergence. Comparing with h Adaptivity	107	(3)
6.3.1 The Shock Problem	107	(2)
6.3.2 The Problem with a Singular Solution	109	(1)
6.4 Discussion of the hp Algorithm	110	(2)
6.5 Algebraic Complexity and Reliability of the Algorithm	112	(2)
6.5.1 Cost of the Linear Solver	113	(1)
6.5.2 Cost of Mesh Optimization	113	(1)
6.5.3 Error Estimation	114	(1)
Exercises	114	(3)
7 Wave Propagation Problems	117	(16)
7.1 Convergence Analysis for Noncoercive Problems	117	(4)
7.1.1 Interpretation with the Projection Operator	120	(1)
7.2 Wave Propagation Problems	121	(4)
7.2.1 Aubin-Nitsche Duality Argument	123	(2)
7.3 Asymptotic Optimality of the Galerkin Method	125	(2)
7.4 Dispersion Error Analysis	127	(4)
Exercises	131	(2)
Part II 2D Elliptic Problems	133	(150)
8 2D Elliptic Boundary-Value Problem	135	(14)
8.1 Classical Formulation	135	(4)
8.1.1 Interface Problems	137	(1)
8.1.2 Regularity of the Solution	138	(1)
8.2 Variational (Weak) Formulation	139	(4)
8.2.1 Interface Problem	142	(1)
8.3 Algebraic Structure of the Variational Formulation	143	(1)
8.4 Equivalence with a Minimization Problem	144	(1)
8.5 Examples from Mechanics and Physics	145	(1)
8.5.1 The Membrane Problem	145	(1)
8.5.2 Torsion of a Shaft	146	(1)
8.5.3 Diffusion–Convection–Reaction Equation	146	(1)
Exercises	146	(3)
9 Sobolev Spaces	149	(14)
9.1 Sobolev Space H¹(Ω)	149	(2)
9.1.1 Distributional Derivatives	149	(2)
9.1.2 Finite-Energy Solutions. Sobolev Space of Order One	151	(1)
9.2 Sobolev Spaces of an Arbitrary Order	151	(6)
9.2.1 Sobolev Spaces of Arbitrary Integer Order	151	(2)
9.2.2 Sobolev Spaces of Arbitrary Real Order. Sloboditskii's Definition	153	(1)
9.2.3 Hörmander's Definition	154	(1)
9.2.4 Interpolation Spaces	155	(2)
9.3 Density and Embedding Theorems	157	(1)
9.4 Trace Theorem	158	(1)
9.5 Well Posedness of the Variational BVP	159	(2)
9.5.1 Continuity of the Bilinear and Linear Forms	159	(1)
9.5.2 Coercivity	159	(1)
9.5.3 The Case with "Pure Neumann" Boundary Conditions	160	(1)
Exercises	161	(2)
10 2D hp Finite Element Method on Regular Meshes	163	(22)
10.1 Quadrilateral Master Element	164	(2)
10.2 Triangular Master Element	166	(3)
10.3 Parametric Element	169	(1)
10.4 Finite Element Space. Construction of Basis Functions	169	(3)
10.4.1 Setting Up the Orientation for Edges	171	(1)
10.5 Calculation of Element Matrices	172	(5)
10.5.1 Computation of the Boundary Terms	174	(1)
10.5.2 Numerical Integration	175	(2)
10.6 Modified Element. Imposing Dirichlet Boundary Conditions	177	(2)
10.7 Postprocessing — Local Access to Element d.o.f	179	(1)
10.8 Projection-Based Interpolation	180	(3)
Exercises	183	(2)
11 2D hp Code	185	(10)
11.1 Getting Started	185	(1)
11.2 Data Structure in FORTRAN 90	186	(2)
11.3 Fundamentals	188	(1)
11.3.1 System Files, IO	188	(1)
11.3.2 The Main Program	188	(1)
11.3.3 Graphics	188	(1)
11.3.4 Quadrature Data	189	(1)
11.4 The Element Routine	189	(1)
11.5 Modified Element. Imposing Dirichlet Boundary Conditions	190	(1)
11.5.1 Imposing Dirichlet Boundary Conditions	191	(1)
11.6 Assignment 4: Assembly of Global Matrices	191	(2)
11.7 The Case with "Pure Neumann" Boundary Conditions	193	(2)
12 Geometric Modeling and Mesh Generation	195	(16)
12.1 Manifold Representation	195	(2)
12.2 Construction of Compatible Parametrizations	197	(4)
12.2.1 Transfinite Interpolation for a Rectangle	200	(1)
12.2.2 Transfinite Interpolation for a Triangle	201	(1)
12.3 Implicit Parametrization of a Rectangle	201	(2)
12.4 Input File Preparation	203	(2)
12.5 Initial Mesh Generation	205	(6)
12.5.1 The Case of "Pure" Neumann Conditions	209	(1)
12.5.2 Defining Profiles	210	(1)
13 The hp Finite Element Method on h-Refined Meshes	211	(16)
13.1 Introduction. The h Refinements	211	(2)
13.2 1-Irregular Mesh Refinement Algorithm	213	(4)
13.3 Data Structure in Fortran 90 (Continued)	217	(2)
13.3.1 The Genealogical Information for Nodes	217	(1)
13.3.2 The Natural Order of Elements	218	(1)
13.4 Constrained Approximation for C° Discretizations	219	(3)
13.4.1 Modified Element	220	(2)
13.5 Reconstructing Element Nodal Connectivities	222	(2)
13.5.1 Definition of the Modified Element	222	(2)
13.6 Determining Neighbors for Midedge Nodes	224	(1)
13.7 Additional Comments	225	(2)
13.7.1 Assembly of Global Matrices. Interfacing with Solvers	225	(1)
13.7.2 Evaluation of Local d.o f	226	(1)
13.7.3 p Refinements	226	(1)
14 Automatic hp Adaptivity in 2D	227	(14)
14.1 The Main Idea	227	(1)
14.2 The 2D hp Algorithm	228	(6)
14.3 Example: L-Shape Domain Problem	234	(2)
14.4 Example: 2D "Shock" Problem	236	(2)
14.5 Additional Remarks	238	(3)
14.5.1 Error Computation in the Case of a Known Solution	238	(1)
14.5.2 Maximum Number of Iterations, Debugging	239	(1)
14.5.3 Systems of Equations and Complex-Valued Problems	239	(1)
14.5.4 Automatic h Adaptivity	239	(2)
15 Examples of Applications	241	(18)
15.1 A "Battery Problem"	241	(3)
15.2 Linear Elasticity	244	(7)
15.3 An Axisymmetric Maxwell Problem	251	(5)
Exercises	256	(3)
16 Exterior Boundary-Value Problems	259	(24)
16.1 Variational Formulation. Infinite Element Discretization	260	(5)
16.1.1 System of Coordinates	261	(1)
16.1.2 Incorporating the Far-Field Pattern. Change of Dependent Variable	262	(3)
16.2 Selection of IE Radial Shape Functions	265	(2)
16.2.1 Same Trial and Test Shape Functions (Bubnov-Galerkin)	265	(1)
16.2.2 Different Trial and Test Shape Functions (Petrov-Galerkin)	266	(1)
16.3 Implementation	267	(3)
16.3.1 Data Structure, Interface with Frontal Solver	267	(1)
16.3.2 Choice of Radial Order N	268	(1)
16.3.3 Calculation of Infinite Element Stiffness Matrix. Routine infel/eleminf	269	(1)
16.4 Calculation of Echo Area	270	(1)
16.4.1 Direct Evaluation Using the IE Solution	270	(1)
16.4.2 Evaluation through Postprocessing	270	(1)
16.5 Numerical Experiments	271	(8)
16.5.1 Evaluation of the Approximation Error	271	(1)
16.5.2 Selection of Radial Shape Functions. Conditioning	271	(2)
16.5.3 Scattering of a Plane Wave on a Rigid Cylinder	273	(2)
16.5.4 Comparison of Bubnov-Galerkin and Petrov-Galerkin Formulations	275	(1)
16.5.5 Scattering of a Plane Wave on a Wedge	275	(2)
16.5.6 Does the Resolution of Singularities Affect the Quality of the Echo Area?	277	(1)
16.5.7 Evaluation of EA	277	(2)
16.6 Comments	279	(1)
Exercises	280	(3)
Part III 2D Maxwell Problems	283	(88)
17 2D Maxwell Equations	285	(16)
17.1 Introduction to Maxwell's Equations	286	(3)
17.1.1 Wave Equation	288	(1)
17.1.2 Time-Harmonic Wave Equation	288	(1)
17.2 Variational Formulation	289	(7)
17.2.1 Nondimensionalization of Maxwell's Equations	294	(2)
Exercises	296	(5)
18 Edge Elements and the de Rham Diagram	301	(22)
18.1 Exact Sequences	301	(8)
18.1.1 Nedelec's Triangular Element of the Second Type	303	(1)
18.1.2 Nedelec's Rectangular Element of the First Type	304	(1)
18.1.3 Nedelec's Triangular Elements of the First Type	305	(3)
18.1.4 Parametric Elements	308	(1)
18.2 Projection-Based Interpolation	309	(5)
18.3 De Rham Diagram	314	(2)
18.4 Shape Functions	316	(5)
18.4.1 Quadrilateral Element	316	(1)
18.4.2 Triangle of the Second Type	317	(2)
18.4.3 Triangle of the First Type	319	(2)
Exercises	321	(2)
19 2D Maxwell Code	323	(14)
19.1 Directories. Data Structure	323	(2)
19.2 The Element Routine	325	(3)
19.3 Constrained Approximation. Modified Element	328	(3)
19.3.1 1D h-Extension Operator	328	(1)
19.3.2 2D h-Extension Operators	329	(1)
19.3.3 Adaptivity	330	(1)
19.3.4 Modified Element. Constrained Approximation	330	(1)
19.3.5 Interface with the Frontal Solver	331	(1)
19.4 Setting Up a Maxwell Problem	331	(3)
19.4.1 Example: Scattering of a Plane Wave on a Screen	332	(2)
Exercises	334	(3)
20 hp Adaptivity for Maxwell Equations	337	(14)
20.1 Projection-Based Interpolation Revisited	338	(3)
20.1.1 Generalized Projection-Based Interpolation	339	(1)
20.1.2 Conflict between the Constrained Nodes and the Commutativity Property	340	(1)
20.2 The hp Mesh Optimization Algorithm	341	(4)
20.3 Example: The Screen Problem	345	(6)
21 Exterior Maxwell Boundary-Value Problems	351	(18)
21.1 Variational Formulation	352	(1)
21.2 Infinite Element Discretization in 3D	353	(4)
21.2.1 Infinite Element Coordinates	354	(2)
21.2.2 Incorporating the Far-Field Pattern	356	(1)
21.2.3 Discretization	357	(1)
21.3 Infinite Element Discretization in 2D	357	(1)
21.3.1 2D IE Coordinates	357	(1)
21.3.2 Incorporating the Far-Field Pattern	358	(1)
21.4 Stability	358	(2)
21.5 Implementation	360	(2)
21.5.1 Automatic hp Adaptivity	361	(1)
21.5.2 Choice of Radial Order N	361	(1)
21.5.3 Evaluation of the Error	362	(1)
21.6 Numerical Experiments	362	(5)
21.6.1 Scattering of a Plane Wave on a PEC Cylinder. Verification of the Code	362	(2)
21.6.2 Scattering of a Plane Wave on a PEC Wedge	364	(1)
21.6.3 Evaluation of RCS	365	(2)
Exercises	367	(2)
22 A Quick Summary and Outlook	369	(2)
Appendix A	371	(4)
Bibliography	375	(14)
Index	389