SMA-HPC '2002 MIT 2-D Integration (from 3-D problems) Reminder Calculating Matrix Elements Panel j i c x Collocation point ( , , i cij panel j Gx xA dS ′′ = ∫ ( , , i c entroii dj Area G x xA ≈ ? One point quadrature Approximation x y z ( int 4 , 1 * 4 i oij c j p Ar A ea Gx x = ≈ ∑ Four point quadrature Approximation ) ) j c ) , j SMA-HPC '2002 MIT Symmetrically Normalized 2-D Problem Quadrature Scheme General n-point formula , i xy n points, n weights 2n parameters -1 1 1-1 y x ( ( 1 11 11 ,, n i i i wf xfx yydx = ?? ∫∫ ∑ . i ) ) i SMA-HPC '2002 MIT Symmetrically Normalized 2-D Problem Quadrature Scheme 2-D Gaussian Quadrature ( ( 11 1 1 1 , , n il i i i l wy pxx yp x = ?? = ∑ ∫∫ l-th order 2-D poly definition Exactness for l-th order polys ( ( , , i l j ij l p xy x yα +≤ = ∑ ( ( ) 1 Number of terms = 2 l + ) ) d ) ) j i ) 2l + SMA-HPC '2002 MIT Symmetrically Normalized 2-D Problem Quadrature Scheme Product Method 1) Get a 1-d formula () ( 1 1 1 1 m d i i wfxfxdx = ? ∫ ∑ . 2) Form a product grid 1 1 d x 1d m x 1d m x 1 1 d x 1 , d i m m x y xx= ) 1 d i 1 , d i SMA-HPC '2002 MIT Symmetrically Normalized 2-D Problem Quadrature Scheme Product Method Continued 3) Determine the weights ( ( 11 1 , 1 1 1 1 , , m d ij i j i wfx xf xydx = ?? ∫ ∑∑ ∫ . 1 , d ij i j w w= Note that n = mxm ) ) 1 m d i = 1 d w SMA-HPC '2002 MIT Symmetrically Normalized 2-D Problem Quadrature Scheme Product Method Theorem Theorem: The product method is exact for all 2-d polys up to order 2m Proof: ( ( 11 11 , 11 1 1 , i m j ij m px y dxdy x y dxdyα +≤ ?? ? ? = ∑ ∫∫ ∫∫ Using the l-th order poly def ) ) j i SMA-HPC '2002 MIT Symmetrically Normalized 2-D Problem Quadrature Scheme Product Theorem Cont. ( ( ) 11 1 1 , 11 1 1 i i j ij ij ij m i j m x y dxdy x dx y dyα +≤ + ≤ ?? ? ? ? = ? ? ∑ ∫∫ ∫ ∫ Using the properties of integration ( 1 , 1 i ij ij m xdx y dyα +≤ ? ? ? ? = ? ? ? ? ? ? ∑ ∫ ) , j α ? ? ? ∑ ) 1 1 j ? ? ? ? ∫ SMA-HPC '2002 MIT Symmetrically Normalized 2-D Problem Quadrature Scheme Product Theorem Cont. Since the 1-d quadrature is exact for polys of order less than 2m ( 1 , 1 i ij ij m xdx y dyα +≤ ? ? ? ? = ? ? ? ? ? ? ∑ ∫ ( ( ( 1 1 1 , 1 m i d d ij k k l l ij m k l w w α +≤ = = ? ? ? ? ? ? ? ? ? ∑ ∑ ) 1 1 j ? ? ? ? ∫ ) ) ) 1 1 m j d d x x ? ? ? ∑ SMA-HPC '2002 MIT Symmetrically Normalized 2-D Problem Quadrature Scheme Product Theorem Cont. Rearranging the sums ( ( ( 1 1 , 1 m i d d ij k k l l ij m k l w w α +≤ = = ? ? ? = ? ? ? ? ? ? ∑ ∑ ( ( ( 1 1 1 , 1 m i d d d k i j k l l i j m ww x xα = + ≤ = ∑∑ ∑ ( 1 1 1 1 , m d d d k m k l l ww p x x = ∑∑ Which proves the theorem ) ) ) 1 1 1 m j d d x x ? ? ? ∑ ) ) ) 1 1 m j d l k = ) 1 1 m d l k = SMA-HPC '2002 MIT 3-D Laplace s Equation Basis Function Approach Calculating Self-Term Panel i i c x Collocation point , 1 i ii pa cnel i x A dS ′ ′ ? = ∫ , 0 i ii c Panel Area A x ? ≈ %(&(' One point quadrature Approximation x y z , is an integrable singularity 1 i ii panel i c x A S ′ ? ′ = ∫ x i c x x d SMA-HPC '2002 MIT 3-D Laplace s Equation Basis Function Approach Calculating Self-Term Tricks of the trade Panel i i c x Collocation point , 1 i ii pa cnel i x A dS ′ ′ ? = ∫ x y z Disk surrounding collocation point , 1 i c ii disk rest of panel A dS dS x x ′ ? ′ = ∫ Disk Integral has singularity but has analytic formula Integrate in two pieces 2 0 1 2 1 i R dk cis dS rdrd R r x π θ ′ ′ = ? ∫ ∫ x of radius R 1 i c x x ′ ? ′ + ∫ 0 x π = ∫ SMA-HPC '2002 MIT 3-D Laplace s Equation Basis Function Approach Calculating Self-Term Other Tricks of the trade Panel i i c x Collocation point Integrand is singular , 1 i ii panel i c x A dS ′ ′ = ? ∫ %(&(' x y z 2) Curve panels can be handled with projection 1) If panel is a flat polygon, analytical formulas exist x