Topic A study on the degenerate scale by
Topic: A study on the degenerate scale by using the fundamental solution with dimensionless argument for 2 D elasticity problems Authors: J. T. Chen(陳正宗) ); Y. T. Lee(李應德) ; J. W. Lee(李家瑋); S. K. Chen(陳聖劻) Date: 2021/01/29 (FRI. ) Place:台大土研大樓 402, NTU 1
Outline Ø Introduction Ø Problem statement Ø Regularization (Enriched fundamental solution) & numerical results Ø Conclusions 2
Outline Ø Introduction Ø Problem statement Ø Regularization (Enriched fundamental solution) & numerical results Ø Conclusions 3
BEM/BIEM (Advantage) To calculate the summation of the arithmetic sequence B 1+2+3+. . . +100=? To calculate the area & volume of the domain Area = ? B Planimeter Area meter 3 D BEM Gauss' theorem Green's identities Somigliana identity Gauss (1777 -1855) 2 D B 1. Reduce the dimension for the integral formula (surface integration to line integration ) 2. Only discrete on the boundary/surface 2. Widely solve the infinite plane problem & crack problem 4
BEM/BIEM (Failure) PDE model(FEM, FVM) Existence unique solution Engineering problems IE model(BEM/BIEM) solution space Non-unique solution (unreasonable result) A mathematical problem for the existence and uniqueness theorem Test function (1) Fundamental sol (2) Green function There is no free lunch when we use BEM/BIEM. Four Parasite must be overcame (1) (2) (3) (4) Degenerate scale Degenerate boundary Spurious eigenvalues Fictitious frequency Range deficiency Rank deficiency Singular matrix 5
Regularization techniques Engineers may not know why numerical instability occurs. Is it a paradox in BEM? Mathematical method Parasite paradise 6
Analytical derivation (Objectivity) From the view of dimensional analysis, it is different result when we employ different unit Nature log Trigonometric function(Sine) Radians Exponential Hankel function Degrees The objectivity should be confirmed, so we must ensure the argument being dimensionle radius arc 7
Our regularization Conventional boundary element method(CBEM) (Original fundamental sol. ) (Rank deficiency/Range deficiency/Singular matrix) Employ a dimensionless augment r/L (Enriched fundamental sol. ) Adaptive : L(revising term) follows the extendable or shrinkable boundary to adjust 8
Outline Ø Introduction Ø Problem statement Ø Regularization (Enriched fundamental solution) & numerical results Ø Conclusions 9
Problem statement Degenerate-scale problem Governing equation (2 D elasticity problem) Integral formula (Single-layer integral operator) Boundary condition (Dirichlet B. C. ) 10
feasibility 2 D elasticity problem Navier equation (Vector) Potential problem Laplace equation (Scalar) Conventional boundary element method(CBEM) complete Adding a rigid body mode Comput. Struct. (2003) complete Fichera’s method IPSE(2014) complete Adding an adaptive characteristic length TWSIAM(2018) present study Conventional boundary element method(CBEM) Adding a rigid body mode Fichera’s method EABE(2016) Adding an adaptive characteristic length ACMFMS(2018) 11
Outline Ø Introduction Ø Problem statement Ø Regularization (Enriched fundamental solution) & numerical results Ø Conclusions 12
The original fundamental solution (Kelvin solution) i=1, 2 j=1, 2 unregularized Degenerate scale 13
Mathematical tool, Degenerate kernel (Circle) On the circular domain: (polar coordinates) Closed-form fundamental solution x rewritten s x Degenerate-form fundamental solution 14
DEGENERATE SCALES IN THE CIRCULAR DOMAIN (unknown boundary densities) (boundary conditions ) substituting the degenerate kernel Ordinary scale: is unique. Degenerate scale: the constant term can’t be determined due to a zero denominator 15
The enriched fundamental solution Because the “characteristic” boundary yields the ill-posed system, we utilized the “characteristic” length to overcome the degenerate scale problem The enriched fundamental solution with dimensionless argument rewritten regularized Depend on the domain For any scale and any material parameter 16
The treatment for our method As mentioned derivation previously, we can derive new constant terms in coefficients Ensuring the argument being dimensionless 1. the geometry parameter is eliminated in discriminant 2. It prevents the material constant to cause a zero denominator in reasonable range. radius diameter perimeter We can avoid the degenerate scale exciting and ensure the unique solution in the BIEM 17
Numerical case(Circular domain) Number of constant elements : 100 Given : New Degenerate scale Numerical (0. 861287) Analytical (0. 860690) Original Degenerate scale Numerical (1. 28400) Analytical (1. 28403) 18
Numerical case(elliptical domain) Number of boundary elements : 100 Given : New Degenerate scale Numerical (0. 52813; 0. 62392) Analytical (0. 52792; 0. 62366) Original Degenerate scale Numerical(0. 78789; 0. 93078) Analytical(0. 78757; 0. 93040) 19
Numerical case(triangular domain) Number of boundary elements : 120 Given : New Degenerate scale Numerical (2. 012) Original Degenerate scale Numerical (3. 00004) 20
Outline Ø Introduction Ø Problem statement Ø Regularization (Enriched fundamental solution) & numerical results Ø Conclusions 21
Summary 1. we clarify the role of an adaptive characteristic length for enriching the deficient range 2. It is different from adding a rigid body mode to shift degenerate scale only 3. For an arbitrary geometry, the perimeter of the domain may be a better consideration 22
END Thank you for listening 23
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