Optimization of elliptic systems: theory and applications
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2006
|
| Edition: | 1. ed. |
| Series: | Springer monographs in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 507 S. |
| ISBN: | 0387272356 9780387272351 9781441920935 |
Staff View
MARC
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| 245 | 1 | 0 | |a Optimization of elliptic systems |b theory and applications |c Pekka Neittaanmaki ; Jürgen Sprekels ; Dan Tiba |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a New York, NY |b Springer |c 2006 | |
| 300 | |a XV, 507 S. | ||
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| 490 | 0 | |a Springer monographs in mathematics | |
| 650 | 4 | |a Elliptisches System - Optimale Kontrolle | |
| 650 | 4 | |a Differential equations, Elliptic |x Numerical solutions | |
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Record in the Search Index
| _version_ | 1804134547595460608 |
|---|---|
| adam_text | Contents
Preface
....................................
v
A Brief Reader s Guide
......................... xi
Chapter
1.
Introductory Topics
.................... 1
1.1
Some General Notions
........................ 1
1.2
Motivating Examples
........................ 3
1.2.1
Cost Functionals
...................... 3
1.2.2
Partial Differential Equations Setting
........... 6
1.2.3
Applications
......................... 14
1.2.4
Variable Domains
..................... 20
Chapter
2.
Existence
........................... 25
2.1
A General Situation
......................... 25
2.2
Special Existence and Uniqueness Results
............. 33
2.2.1
Second-Order Problems
.................. 33
2.2.2
Fourth-Order Problems
................... 39
2.3
Variable Domains
.......................... 44
2.3.1
Some Examples
....................... 45
2.3.2
General Dirichlet Problems
................ 53
2.3.3
Neumann and Mixed Boundary Conditions
........ 64
2.3.4
Partial Extensions
..................... 75
Chapter
3.
Optimality Conditions
................... 83
3.1
Abstract Approaches
......................... 83
3.1.1
The Convex Case. Subdifferential Calculus
........ 83
3.1.2
The Convex Case. Mathematical Programming
..... 88
3.1.3
The Differentiable Case
................... 92
3.1.3.1
Singular Control Problems
........... 96
xiii
χ^ν
Concents
3.2
Penalization
.............................
Ю2
3.2.1
The Standard Approach
.................. 103
3.2.2
Penalization of the State Equation
............ 113
3.2.3 Semilinear
Equations and Exact Penalization
....... 127
3.3
Control of Variational Inequalities
................. 143
3.3.1
An Abstract Result
..................... 143
3.3.2 Semilinear
Variational Inequalities
............. 148
3.3.3
State Constraints and Penalization of the Equation
. . . 159
3.4
Thickness Optimization for Plates
................. 168
3.4.1
Simply Supported Plates
.................. 168
3.4.2
Clamped Plates and Control Variational Methods
.... 175
3.4.2.1
Penalization and Variational Inequalities
. . . 183
Chapter
4.
Discretization
........................ 193
4.1
Finite Element Approximation of Elliptic Equations
....... 193
4.1.1
The Finite Element Method
................ 194
4.1.2
Error Estimates for the FE Equations
........... 198
4.2
Error Estimates in the Finite Element Discretization of Control
Problems
.......................... 206
4.3
Semidiscretization
.......................... 220
4.4
Optimal Control Problems Governed by Elliptic Variational
Inequalities
......................... 226
4.4.1
The Ritz-Galerkin Approximation
............. 228
4.5
Error Estimates in the Discretization of Control Problems with
Nonlinear State Equation
................. 238
Chapter
5.
Unknown Domains
..................... 249
5.1
Free Boundary Problems
...................... 249
5.1.1
The Dam Problem
..................... 250
5.1.2
Free Boundary Problems and Optimal Design
...... 254
5.1.3
Shape Optimization of Systems with Free Boundaries
. . 258
5.1.4
Controllability of the Coincidence Set
........... 267
5.2
Direct Approaches
.......................... 278
5.2.1
An Algorithm of
Céa,
Gioan, and Michel
......... 278
5.2.2
Characteristic Functions
.................. 284
5.2.2.1
Structural Material Optimization
....... 285
Contents xv
5.2.2.2 Bang-Bang Controls
and Characteristic
Functions
.................... 294
5.2.3
Controllability and Fictitious Domains Approaches
. . . 297
5.2.3.1
Boundary Controllability
............ 297
5.2.3.2
Distributed Controls
.............. 310
5.3
Domain Variations
.......................... 323
5.3.1
Classical Approaches
.................... 323
5.3.2
Topological Asymptotics
.................. 334
Chapter
6.
Optimization of Curved Mechanical Systems
.... 341
6.1
Kirchhoff-Love Arches
........................ 342
6.1.1
Application of the Control Variational Method
...... 343
6.1.2
Optimization of Nonsmooth Arches
............ 351
6.1.3
Variational Inequalities for Arches
............. 369
6.2
General Three-Dimensional Curved Rods
............. 371
6.2.1
A Local Frame Under Low Differentiability Assumptions
372
6.2.2
A Generalized Naghdi-Type Model
............ 374
6.2.3
Shape Optimization
..................... 389
6.3
Applications to Shells
........................ 406
6.3.1
A Generalized Naghdi Shell Model
............ 406
6.3.2
Proof of Coercivity
..................... 413
6.3.3
Shell Optimal Design
.................... 422
Appendix
1.
Convex Mappings and Monotone Operators
.... 437
Appendix
2.
Elliptic Equations and Variational Inequalities
. . 451
Appendix
3.
Domain Convergence
.................. 459
Bibliography
................................ 475
Index
..................................... 495
Notation
................................... 505
|
| adam_txt |
Contents
Preface
.
v
A Brief Reader's Guide
. xi
Chapter
1.
Introductory Topics
. 1
1.1
Some General Notions
. 1
1.2
Motivating Examples
. 3
1.2.1
Cost Functionals
. 3
1.2.2
Partial Differential Equations Setting
. 6
1.2.3
Applications
. 14
1.2.4
Variable Domains
. 20
Chapter
2.
Existence
. 25
2.1
A General Situation
. 25
2.2
Special Existence and Uniqueness Results
. 33
2.2.1
Second-Order Problems
. 33
2.2.2
Fourth-Order Problems
. 39
2.3
Variable Domains
. 44
2.3.1
Some Examples
. 45
2.3.2
General Dirichlet Problems
. 53
2.3.3
Neumann and Mixed Boundary Conditions
. 64
2.3.4
Partial Extensions
. 75
Chapter
3.
Optimality Conditions
. 83
3.1
Abstract Approaches
. 83
3.1.1
The Convex Case. Subdifferential Calculus
. 83
3.1.2
The Convex Case. Mathematical Programming
. 88
3.1.3
The Differentiable Case
. 92
3.1.3.1
Singular Control Problems
. 96
xiii
χ^ν
Concents
3.2
Penalization
.
Ю2
3.2.1
The Standard Approach
. 103
3.2.2
Penalization of the State Equation
. 113
3.2.3 Semilinear
Equations and Exact Penalization
. 127
3.3
Control of Variational Inequalities
. 143
3.3.1
An Abstract Result
. 143
3.3.2 Semilinear
Variational Inequalities
. 148
3.3.3
State Constraints and Penalization of the Equation
. . . 159
3.4
Thickness Optimization for Plates
. 168
3.4.1
Simply Supported Plates
. 168
3.4.2
Clamped Plates and Control Variational Methods
. 175
3.4.2.1
Penalization and Variational Inequalities
. . . 183
Chapter
4.
Discretization
. 193
4.1
Finite Element Approximation of Elliptic Equations
. 193
4.1.1
The Finite Element Method
. 194
4.1.2
Error Estimates for the FE Equations
. 198
4.2
Error Estimates in the Finite Element Discretization of Control
Problems
. 206
4.3
Semidiscretization
. 220
4.4
Optimal Control Problems Governed by Elliptic Variational
Inequalities
. 226
4.4.1
The Ritz-Galerkin Approximation
. 228
4.5
Error Estimates in the Discretization of Control Problems with
Nonlinear State Equation
. 238
Chapter
5.
Unknown Domains
. 249
5.1
Free Boundary Problems
. 249
5.1.1
The Dam Problem
. 250
5.1.2
Free Boundary Problems and Optimal Design
. 254
5.1.3
Shape Optimization of Systems with Free Boundaries
. . 258
5.1.4
Controllability of the Coincidence Set
. 267
5.2
Direct Approaches
. 278
5.2.1
An Algorithm of
Céa,
Gioan, and Michel
. 278
5.2.2
Characteristic Functions
. 284
5.2.2.1
Structural Material Optimization
. 285
Contents xv
5.2.2.2 Bang-Bang Controls
and Characteristic
Functions
. 294
5.2.3
Controllability and Fictitious Domains Approaches
. . . 297
5.2.3.1
Boundary Controllability
. 297
5.2.3.2
Distributed Controls
. 310
5.3
Domain Variations
. 323
5.3.1
Classical Approaches
. 323
5.3.2
Topological Asymptotics
. 334
Chapter
6.
Optimization of Curved Mechanical Systems
. 341
6.1
Kirchhoff-Love Arches
. 342
6.1.1
Application of the Control Variational Method
. 343
6.1.2
Optimization of Nonsmooth Arches
. 351
6.1.3
Variational Inequalities for Arches
. 369
6.2
General Three-Dimensional Curved Rods
. 371
6.2.1
A Local Frame Under Low Differentiability Assumptions
372
6.2.2
A Generalized Naghdi-Type Model
. 374
6.2.3
Shape Optimization
. 389
6.3
Applications to Shells
. 406
6.3.1
A Generalized Naghdi Shell Model
. 406
6.3.2
Proof of Coercivity
. 413
6.3.3
Shell Optimal Design
. 422
Appendix
1.
Convex Mappings and Monotone Operators
. 437
Appendix
2.
Elliptic Equations and Variational Inequalities
. . 451
Appendix
3.
Domain Convergence
. 459
Bibliography
. 475
Index
. 495
Notation
. 505 |
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| id | DE-604.BV020840337 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T13:16:17Z |
| indexdate | 2024-07-09T20:26:21Z |
| institution | BVB |
| isbn | 0387272356 9780387272351 9781441920935 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014162229 |
| oclc_num | 255321688 |
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| physical | XV, 507 S. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer monographs in mathematics |
| spelling | Neittaanmäki, Pekka 1951- Verfasser (DE-588)132647702 aut Optimization of elliptic systems theory and applications Pekka Neittaanmaki ; Jürgen Sprekels ; Dan Tiba 1. ed. New York, NY Springer 2006 XV, 507 S. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Elliptisches System - Optimale Kontrolle Differential equations, Elliptic Numerical solutions Optimierung (DE-588)4043664-0 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Elliptisches System (DE-588)4121184-4 gnd rswk-swf Optimierung (DE-588)4043664-0 s Elliptisches System (DE-588)4121184-4 s DE-604 Elliptische Differentialgleichung (DE-588)4014485-9 s 1\p DE-604 Sprekels, Jürgen 1948- Verfasser (DE-588)108705323 aut Tiba, Dan Verfasser aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014162229&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Neittaanmäki, Pekka 1951- Sprekels, Jürgen 1948- Tiba, Dan Optimization of elliptic systems theory and applications Elliptisches System - Optimale Kontrolle Differential equations, Elliptic Numerical solutions Optimierung (DE-588)4043664-0 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Elliptisches System (DE-588)4121184-4 gnd |
| subject_GND | (DE-588)4043664-0 (DE-588)4014485-9 (DE-588)4121184-4 |
| title | Optimization of elliptic systems theory and applications |
| title_auth | Optimization of elliptic systems theory and applications |
| title_exact_search | Optimization of elliptic systems theory and applications |
| title_exact_search_txtP | Optimization of elliptic systems theory and applications |
| title_full | Optimization of elliptic systems theory and applications Pekka Neittaanmaki ; Jürgen Sprekels ; Dan Tiba |
| title_fullStr | Optimization of elliptic systems theory and applications Pekka Neittaanmaki ; Jürgen Sprekels ; Dan Tiba |
| title_full_unstemmed | Optimization of elliptic systems theory and applications Pekka Neittaanmaki ; Jürgen Sprekels ; Dan Tiba |
| title_short | Optimization of elliptic systems |
| title_sort | optimization of elliptic systems theory and applications |
| title_sub | theory and applications |
| topic | Elliptisches System - Optimale Kontrolle Differential equations, Elliptic Numerical solutions Optimierung (DE-588)4043664-0 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Elliptisches System (DE-588)4121184-4 gnd |
| topic_facet | Elliptisches System - Optimale Kontrolle Differential equations, Elliptic Numerical solutions Optimierung Elliptische Differentialgleichung Elliptisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014162229&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT neittaanmakipekka optimizationofellipticsystemstheoryandapplications AT sprekelsjurgen optimizationofellipticsystemstheoryandapplications AT tibadan optimizationofellipticsystemstheoryandapplications |