ON DISCRETE SHAPE GRADIENTS OF BOUNDARY TYPE FOR PDE-CONSTRAINED SHAPE OPTIMIZATION

doi:10.1137/20M1323898

	ON DISCRETE SHAPE GRADIENTS OF BOUNDARY TYPE FOR PDE-CONSTRAINED SHAPE OPTIMIZATION
	Gong, Wei 3,4; Zhu, Shengfeng 1,2
刊名	SIAM JOURNAL ON NUMERICAL ANALYSIS
	2021
卷号	59 期号:3 页码:1510-1541
关键词	shape optimization shape gradient boundary formulation boundary correction a priori error estimate finite element
ISSN号	0036-1429
DOI	10.1137/20M1323898
英文摘要	Shape gradients have been widely used in numerical shape gradient descent algorithms for shape optimization. The two types of shape gradients, i.e., the distributed one and the boundary type, are equivalent at the continuous level but exhibit different numerical behaviors after finite element discretization. To be more specific, the boundary type shape gradient is more popular in practice due to its concise formulation and convenience in combining with shape optimization algorithms but has lower numerical accuracy. In this paper we provide a simple yet useful boundary correction for the normal derivatives of the state and adjoint equations, motivated by their continuous variational forms, to increase the accuracy and possible effectiveness of the boundary shape gradient in PDE-constrained shape optimization. We consider particularly the state equation with Dirichlet boundary conditions and provide a preliminary error estimate for the correction. Numerical results show that the corrected boundary type shape gradient has comparable accuracy to that of the distributed one. Moreover, we give a theoretical explanation for the comparable numerical accuracy of the boundary type shape gradient with that of the distributed shape gradient for Neumann boundary value problems.
资助项目	Strategic Priority Research Program of Chinese Academy of Sciences[XDB 41000000] ; National Key Basic Research Program[2018YFB0704304] ; National Natural Science Foundation of China[11671391] ; National Natural Science Foundation of China[12071149] ; Science and Technology Commission of Shanghai Municipality[18dz2271000] ; Natural Science Foundation of Shanghai[19ZR1414100] ; Fundamental Research Funds for the Central Universities
WOS研究方向	Mathematics
语种	英语
出版者	SIAM PUBLICATIONS
WOS记录号	WOS:000674273400014
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/59083]
专题	中国科学院数学与系统科学研究院
通讯作者	Gong, Wei
作者单位	1.East China Normal Univ, Shanghai Key Lab Pure Math & Math Practice, Shanghai 200241, Peoples R China 2.East China Normal Univ, Sch Math Sci, Shanghai 200241, Peoples R China 3.Chinese Acad Sci, Acad Math & Syst Sci, Natl Ctr Math & Interdisciplinary Sci, Beijing 100190, Peoples R China 4.Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math, State Key Lab Sci & Engn Comp, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Gong, Wei,Zhu, Shengfeng. ON DISCRETE SHAPE GRADIENTS OF BOUNDARY TYPE FOR PDE-CONSTRAINED SHAPE OPTIMIZATION[J]. SIAM JOURNAL ON NUMERICAL ANALYSIS,2021,59(3):1510-1541.
APA	Gong, Wei,&Zhu, Shengfeng.(2021).ON DISCRETE SHAPE GRADIENTS OF BOUNDARY TYPE FOR PDE-CONSTRAINED SHAPE OPTIMIZATION.SIAM JOURNAL ON NUMERICAL ANALYSIS,59(3),1510-1541.
MLA	Gong, Wei,et al."ON DISCRETE SHAPE GRADIENTS OF BOUNDARY TYPE FOR PDE-CONSTRAINED SHAPE OPTIMIZATION".SIAM JOURNAL ON NUMERICAL ANALYSIS 59.3(2021):1510-1541.