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jghoskinsjghoskins
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axisymmetric laplace kernels
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‎chunkie/+chnk/+axissymlap2d/gaus_agm.m‎

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function [rk0,re0] = gaus_agm(x)
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eps = 1E-15;
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a = sqrt(2./(x+1));
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delt = 1./sqrt(1-a.*a);
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aa0 = delt + sqrt(delt.*delt-1);
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bb0 = 1./(delt+sqrt(delt.*delt-1));
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a0 = ones(size(delt));
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b0 = 1./delt;
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fact = ((a0+b0)/2).^2;
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for i=1:1000
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a1 = (a0+b0)/2;
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b1 = sqrt(a0.*b0);
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aa1 = (aa0+bb0)/2;
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bb1 = sqrt(aa0.*bb0);
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a0 = a1;
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b0 = b1;
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aa0 = aa1;
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bb0 = bb1;
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c0 = (a1-b1)/2;
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fact = fact-(c0.*c0)*2^(i);
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drel = abs(a0-b0)./abs(a0);
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drel2= abs(aa0-bb0)./abs(aa0);
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if (max(drel+drel2) <2*eps)
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break
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end
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end
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rk0 = pi./(2*aa0.*sqrt(1-a.*a));
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re0 = pi*fact./(2*a0);
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end

‎chunkie/+chnk/+axissymlap2d/gfunc.m‎

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function [gval,gdz,gdr,gdrp] = gfunc(r,rp,dr,z,zp,dz)
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domega3 = 2*pi;
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n = 3;
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t = (dz.^2+dr.^2)./(2.*r.*rp);
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[q0,q1,q0d] = chnk.axissymlap2d.qleg_half(t);
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gval = domega3*(rp./r).^((n-2)/2).*q0;
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gdz =-domega3*(rp./r).^((n-2)/2).*q0d ...
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./(rp.*r).*dz;
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rfac = -r*(n-2)/2.*q0+(-(1+t).*r+rp).*q0d;
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gdrp = -domega3*(rp./r).^((n-2)/2)./(rp.*r) ...
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.*rfac;
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rfac = -rp*(n-2)/2.*q0+(-(1+t).*rp+r).*q0d;
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gdr = -domega3*(rp./r).^((n-2)/2)./(rp.*r) ...
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.*rfac;
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end

‎chunkie/+chnk/+axissymlap2d/gfunc_on_axis.m‎

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function [gval,gdz,gdr,gdrp] = gfunc_on_axis(r,rp,dr,z,zp,dz)
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domega3 = 2*pi;
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n = 3;
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t = (dz.^2+dr.^2)./(2.*r.*rp);
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[q0,q1,q0d] = chnk.axissymlap2d.qleg_half(t);
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gval = domega3*(rp./r).^((n-2)/2).*q0;
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gdz =-domega3*(rp./r).^((n-2)/2).*q0d ...
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./(rp.*r).*dz;
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rfac = -r*(n-2)/2.*q0+(-(1+t).*r+rp).*q0d;
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gdrp = -domega3*(rp./r).^((n-2)/2)./(rp.*r) ...
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.*rfac;
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rfac = -rp*(n-2)/2.*q0+(-(1+t).*rp+r).*q0d;
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gdr = -domega3*(rp./r).^((n-2)/2)./(rp.*r) ...
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.*rfac;
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end

‎chunkie/+chnk/+axissymlap2d/green.m‎

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function [val, grad] = green(src, targ, origin)
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%CHNK.AXISSYMHELM2D.GREEN evaluate the Laplace green's function
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% for the given sources and targets.
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%
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% Note: that the first coordinate is r, and the second z.
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% The code relies on precomputed tables and hence loops are required for
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% computing various pairwise interactions.
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% Finally, the code is not efficient in the sense that val, grad, hess
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% are always internally computed independent of nargout
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%
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% Since the kernels are not translationally invariant in r, the size
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% of the gradient is 3, for d/dr, d/dr', and d/dz
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%
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[~, ns] = size(src);
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[~, nt] = size(targ);
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gtmp = zeros(nt, ns, 3);
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rt = repmat(targ(1,:).',1,ns);
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rs = repmat(src(1,:),nt,1);
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dz = repmat(src(2,:),nt,1)-repmat(targ(2,:).',1,ns);
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r = (rt + origin(1));
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rp = (rs + origin(1));
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dr = (rs-rt);
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z = zeros(size(rt));
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zp = zeros(size(rt));
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[g,gdz,gdr,gdrp] = chnk.axissymlap2d.gfunc(r,rp,dr,z,zp,dz);
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val = g;
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gtmp(:,:,1) = gdr;
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gtmp(:,:,2) = gdrp;
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gtmp(:,:,3) = gdz;
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grad = gtmp;
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‎chunkie/+chnk/+axissymlap2d/kern.m‎

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function submat = kern(srcinfo, targinfo, origin, type)
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%CHNK.AXISSYMLAP2D.KERN axissymmetric Laplace layer potential kernels in 2D
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%
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% Syntax: submat = chnk.axissymlap2d.kern(srcinfo,targingo,type)
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%
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% Let x be targets and y be sources for these formulas, with
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% n_x and n_y the corresponding unit normals at those points
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% (if defined). Note that the normal information is obtained
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% by taking the perpendicular to the provided tangential deriviative
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% info and normalizing
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%
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% Here the first and second components correspond to the r and z
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% coordinates respectively.
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%
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% Kernels based on G(x,y) = \int_{0}^{\pi} 1/(d(t)) \, dt \,
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% where d(t) = \sqrt(r^2 + r'^2 - 2rr' \cos(t) + (z-z')^2) with
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% x = (r,z), and y = (r',z')
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%
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% D(x,y) = \nabla_{n_y} G(x,y)
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% S(x,y) = G(x,y)
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% S'(x,y) = \nabla_{n_x} G(x,y)
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% D'(x,y) = \nabla_{n_x} \nabla_{n_y} G(x,y)
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%
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% Input:
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% srcinfo - description of sources in ptinfo struct format, i.e.
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% ptinfo.r - positions (2,:) array
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% ptinfo.d - first derivative in underlying
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% parameterization (2,:)
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% ptinfo.d2 - second derivative in underlying
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% parameterization (2,:)
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% targinfo - description of targets in ptinfo struct format,
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% if info not relevant (d/d2) it doesn't need to
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% be provided. sprime requires tangent info in
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% targinfo.d
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% type - string, determines kernel type
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% type == 'd', double layer kernel D
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% type == 's', single layer kernel S
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% type == 'sprime', normal derivative of single
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% layer S'
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%
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%
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% Output:
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% submat - the evaluation of the selected kernel for the
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% provided sources and targets. the number of
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% rows equals the number of targets and the
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% number of columns equals the number of sources
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%
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%
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src = srcinfo.r;
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targ = targinfo.r;
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[~, ns] = size(src);
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[~, nt] = size(targ);
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if strcmpi(type, 'd')
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srcnorm = srcinfo.n;
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[~, grad] = chnk.axissymlap2d.green(src, targ, origin);
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nx = repmat(srcnorm(1,:), nt, 1);
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ny = repmat(srcnorm(2,:), nt, 1);
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% Due to lack of translation invariance in r, no sign flip needed,
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% as gradient is computed with repsect to r'
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submat = (grad(:,:,2).*nx + grad(:,:,3).*ny);
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end
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if strcmpi(type, 'sprime')
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targnorm = targinfo.n;
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[~, grad] = chnk.axissymlap2d.green(src, targ, origin);
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nx = repmat((targnorm(1,:)).',1,ns);
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ny = repmat((targnorm(2,:)).',1,ns);
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submat = (grad(:,:,1).*nx - grad(:,:,3).*ny);
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end
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if strcmpi(type, 's')
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submat = chnk.axissymlap2d.green(src, targ, origin);
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end
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‎chunkie/+chnk/+axissymlap2d/qeval0_far.m‎

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function [q0] = qeval0_far(x)
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coefs = ...
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[2.2214414690791831235E0,
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0,
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0.41652027545234683566E0,
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0,
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0.22778452563800217575E0,
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0,
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0.15660186137612649583E0,
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0,
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0.11928657409509635424E0];
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t = 1./x;
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t0 = 1./sqrt(x);
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q0 = zeros(size(t));
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for i=1:9
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q0 = q0 + t0*coefs(i);
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t0 = t0.*t;
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end
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end

‎chunkie/+chnk/+axissymlap2d/qeval0_near.m‎

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function [q0] = qeval0_near(t)
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coefs = ...
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[1.7328679513998632735E0,
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-0.5000000000000000000E0,
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-0.09160849392498290919E0,
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0.062500000000000000000E0,
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0.019905513916401443210E0,
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-0.017578125000000000000E0,
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-0.006097834693194945559E0,
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0.006103515625000000000E0,
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0.0021674343381175963469E0,
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-0.0023365020751953125000E0,
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-0.0008357538695841108955E0,
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0.0009462833404541015625E0,
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0.0003390851133264318725E0,
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-0.00039757043123245239258E0,
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-0.00014242013770157621830E0,
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0.00017140153795480728149E0,
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0.00006133159221782055041E0,
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-0.00007532294148404616863E0];
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b = log(t);
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v = 1;
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q0 = zeros(size(t));
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for i=1:9
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q0 = q0 + v*coefs(2*i-1)+v.*b*coefs(2*i);
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v = v.*t;
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end
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end

‎chunkie/+chnk/+axissymlap2d/qeval0der_near.m‎

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function [q0d] = qeval0der_near(t)
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coefs = ...
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[1.7328679513998632735d0,
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-0.5000000000000000000d0,
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-0.09160849392498290919d0,
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0.062500000000000000000d0,
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0.019905513916401443210d0,
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-0.017578125000000000000d0,
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-0.006097834693194945559d0,
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0.006103515625000000000d0,
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0.0021674343381175963469d0,
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-0.0023365020751953125000d0,
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-0.0008357538695841108955d0,
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0.0009462833404541015625d0,
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0.0003390851133264318725d0,
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-0.00039757043123245239258d0,
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-0.00014242013770157621830d0,
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0.00017140153795480728149d0,
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0.00006133159221782055041d0,
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-0.00007532294148404616863d0];
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b = log(t);
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v = 1;
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q0d = coefs(2)./t;
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for i=2:9
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q0d = q0d + (i-1)* ...
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(v*coefs(2*i-1)+v.*b*coefs(2*i)) ...
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+v*coefs(2*i);
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v = v.*t;
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end
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end

‎chunkie/+chnk/+axissymlap2d/qeval1_far.m‎

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function [q1] = qeval1_far(x)
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coefs = ...
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[0.55536036726979578088E0,
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0,
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0.26032517215771677229E0,
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0,
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0.17083839422850163181E0,
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0,
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0.12723901236810277786E0,
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0,
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0.10139358798083190111E0];
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t = 1./x;
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t0 = 1./sqrt(x).^3;
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q1 = zeros(size(t));
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for i=1:9
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q1 = q1 + t0*coefs(i);
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t0 = t0.*t;
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end
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end

‎chunkie/+chnk/+axissymlap2d/qeval1_near.m‎

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function [q1] = qeval1_near(t)
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coefs = ...
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[-0.2671320486001367265d0,
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-0.5000000000000000000d0,
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0.5248254817749487276d0,
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-0.18750000000000000000d0,
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-0.04098835652733573868d0,
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0.029296875000000000000d0,
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0.009513531070472923783d0,
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-0.008544921875000000000d0,
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-0.0029774361551467310174d0,
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0.0030040740966796875000d0,
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0.0010682069932178195667d0,
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-0.0011565685272216796875d0,
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-0.0004138797762860294822d0,
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0.00046985596418380737305d0,
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0.00016838776925222835322d0,
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-0.00019777100533246994019d0,
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-0.00007084821236213522235d0,
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0.00008536600034858565778d0];
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b = log(t);
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v = 1;
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q1 = zeros(size(t));
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for i=1:9
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q1 = q1 + v*coefs(2*i-1)+v.*b*coefs(2*i);
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v = v.*t;
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end
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end

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