Jia-Wei Lee, Chi-Feng Nien & Jeng-Tzong Chen  Symposium of the International Association for Boundary Element Methods 2018 Combination of the CHIEF and the self-regularization technique for solving 2D exterior Helmholtz equations with fictitious frequencies in the indirect BEM and MFS Jia-Wei Lee, Chi-Feng Nien & Jeng-Tzong Chen Date: June 26, 2018 Time: 09:35~10:00 Place: University Paris 6 2018/06/28

Outline Introduction Problem statement Present approach Numerical results Conclusions 2018/06/28

Outline Introduction Problem statement Present approach Numerical results Conclusions 2018/06/28

Engineering problems Platform (Offshore structure) Exterior problem Ringing, Rodney Eatock Taylor BHP Billiton's giant Atlantis oilfield in the deepwater Gulf of Mexico 2018/06/28

Motivation Exterior problem FEM BEM Artificial boundary condition Only boundary element DtN boundary condition Nonreflection boundary condition (NCTS, 2013, Summer Course) 2018/06/28

Unique solution->Paradise No solution->Parasite well-posed ill-posed Unique solution->Paradise No solution->Parasite Infinite domain Paradise Model Parasite Black-box 2018/06/28

Physical and numerical resonance Physical resonance Fictitious frequency (BEM/BIEM) t(a,0) 2018/06/28

Rank-deficency problems Influence matrix Spurious eigenvalue (Interior) Fictitious frequency (Exterior) t(a,0) 2018/06/28

Alleviation for rank-deficency problems Spurious eigenvalue (Interior) Fictitious frequency (Exterior) t(a,0) (1) CHIEF method (Schenck, JASA , 1968) Additional constraint (CHIEF point) (2) Burton and Miller method (Burton and Miller, PRS , 1971) (3) SVD updating term technique (Chen et al., JSV, 2002) 2018/06/28

Mixed potential method Four regularization techniques   Direct BEM indirect BEM (MFS) Burton & Miller method or Mixed potential method CHIEF method Present approach No null-field equation Null-field equation Self-regularization method + CHIEF Extra point 2018/06/28

Outline Introduction Problem statement Present approach Numerical results Conclusions 2018/06/28

Problem statement Exact solution: Domain Governing equation: Boundary condition: Exact solution: 2018/06/28

Indirect BEM Indirect BIE Domain Fundamental solution Boundary Matching the boundary condition Discretization 2018/06/28

Spectrum of minimum singular value single root double root 2018/06/28

Rank deficiency Domain base Range base Operator How to represent for Domain base Range base Operator How to represent for of ?? 2018/06/28

Outline Introduction Problem statement Present approach Numerical results Conclusions 2018/06/28

Self-regularization technique Fichera’s method for the problem of degenerate scale constraint equation When the size is at the degenerate scale, the Fichera’s method can obtain the unique solution 2018/06/28

Self-regularization technique SVD has no component If Self-regularization technique by using the SVD 2018/06/28

Spectrum of U by using the self-regularization method 2.409 3.838 5.144 5.529 6.390 7.027 7.599 8.431 8.668 8.783 9.776 9.947 2018/06/28

A new point of view of CHIEF Exterior problem governed by the Helmholtz equation Direct BEM Fictitious frequency Null-field equation is a CHIEF equation Relation between CHIEF and self-reg. method Risk of failure points Domain Indirect BEM Fictitious frequency No null-field equation How to find CHIEF equation ? Constraint equation ? Failure points ? G. E. B. Cs. or 2018/06/28 A self-regularization method

Self-regularization technique Range deficiency Discrete system How to determine the field Self-regularization ??? Boundary data 2018/06/28

Self-regularization+CHIEF idea Range deficiency Continuous system Extra source point 2018/06/28

Analytical derivation of the present indirect BIE Double root Degenerate kernel Fourier series 2018/06/28

Analytical derivation of the present indirect BIE Double root The failure positions are the same with the CHIEF method in the direct BEM 2018/06/28

Present idea for MFS Constraint equations Ordinary MFS: Present MFS: MFS source point Extra source point Constraint equations Single root Double root 2018/06/28

Outline Introduction Problem statement Present approach Numerical results Conclusions 2018/06/28

Numerical examples Case 1 Case 2 Case 3 Case 4 2018/06/28

Bar chart of unitary vectors CHIEF 2018/06/28

Contour plot Case 1 2018/06/28

Contour plot Case 1 2018/06/28

Contour plot Case 2 2018/06/28

Contour plot Case 2 2018/06/28

Bar chart of unitary vectors CHIEF 2018/06/28

Contour plot Case 3 2018/06/28

Contour plot Case 4 2018/06/28

Contour plot Case 4 Failure points 2018/06/28 2017/07/11

Numerical examples Case 4 Exact solution Ordinary MFS The present MFS with failure points The present MFS with suitable points 2018/06/28

Outline Introduction Problem statement Present approach Numerical results Conclusions 2018/06/28

Conclusions The bordered matrix obtained by the self-regularization technique is invertible for all value of wavenumber. The problem of fictitious frequencies can be alleviated by adding extra fundamental solutions in the direct BEM and MFS. The present approach can fill in the gap that there is no CHIEF method in the indirect BEM and MFS. The criterion of failure points of the present idea is the same with the CHIEF method. 2018/06/28

The end Thanks for your kind attentions 2018/06/28

Welcome to visit the web site of MSVLAB/NTOU The end Thanks for your kind attentions 勞請各位老師與前輩多多指教 Welcome to visit the web site of MSVLAB/NTOU http://msvlab.hre.ntou.edu.tw/ 2018/06/28

Contour plot Case 3 Failure point 2018/06/28 2017/07/11

目前需要克服的問題 COLY(NELM+1)=0.9*DCOS(Pi/(14.)) COLX(NELM+1)=0.9*DSIN(Pi/(14.)) COLX(NELM+2)=0.9 COLY(NELM+2)=0 2017/07/11 2018/06/28

Physical and numerical resonance Physical resonance Fictitious frequency (BEM/BIEM) Present t(a,0) 2017/07/11 2018/06/28

Self-regularization+CHIEF method 怎麼進場 自救 無法表現 可表現 2018/06/28

Analytical derivation of 魯蛋版 indirect BIE 2018/06/28