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author | cdf |
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date | Mon, 08 Jun 2015 08:51:06 +0000 |
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## Copyright (C) 2006-2014 Carlo de Falco, Massimiliano Culpo ## ## This file is part of: ## BIM - Diffusion Advection Reaction PDE Solver ## ## BIM is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 2 of the License, or ## (at your option) any later version. ## ## BIM is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with BIM; If not, see <http://www.gnu.org/licenses/>. ## ## author: Carlo de Falco <cdf _AT_ users.sourceforge.net> ## author: Massimiliano Culpo <culpo _AT_ users.sourceforge.net> ## author: Matteo porro <meoo85 _AT_ users.sourceforge.net> ## author: Emanuela Abbate <emanuela.abbate _AT_ mail.polimi.it> ## -*- texinfo -*- ## @deftypefn {Function File} @ ## {[@var{A}]} = @ ## bim2a_axisymmetric_advection_diffusion(@var{mesh},@var{alpha},@var{gamma},@var{eta},@var{beta}) ## ## Build the Scharfetter-Gummel stabilized stiffness matrix for a ## diffusion-advection problem in cylindrical coordinates with axisymmetric ## configuration. Rotational symmetry is assumed with respect to be the vertical ## axis r=0. Only plane geometries that DO NOT intersect the symmetry axis ## are admitted. ## ##@example ##@group ## | ____ _|____ ## | | \ \ | | ## z | | \ OK \| | NO! ## | |______\ |\___| ## | r | #@end group #@end example ## ## The equation taken into account is: ## ## 1/r * d(r * Fr)/dr + dFz/dz = f ## ## with ## ## F = [Fr, Fz]' = - @var{alpha} * @var{gamma} ( @var{eta} grad (u) - @var{beta} u ) ## ## where @var{alpha} is an element-wise constant scalar function, ## @var{eta} and @var{gamma} are piecewise linear conforming scalar ## functions, @var{beta} is an element-wise constant vector function. ## ## Instead of passing the vector field @var{beta} directly, one can pass ## a piecewise linear conforming scalar function @var{phi} as the last ## input. In such case @var{beta} = grad @var{phi} is assumed. ## ## If @var{phi} is a single scalar value @var{beta} is assumed to be 0 ## in the whole domain. ## ## @seealso{bim2a_axisymmetric_rhs, bim2a_axisymmetric_reaction, ## bim2a_advection_diffusion, bim2c_mesh_properties} ## @end deftypefn function [A] = bim2a_axisymmetric_advection_diffusion (mesh, alpha, gamma, eta, beta) ## Check input if nargin != 5 error("bim2a_axisymmetric_advection_diffusion: wrong number of input parameters."); elseif !(isstruct(mesh) && isfield(mesh,"p") && isfield (mesh,"t") && isfield(mesh,"e")) error("bim2a_axisymmetric_advection_diffusion: first input is not a valid mesh structure."); elseif !(all(mesh.p(1,:) >= 0) || all(mesh.p(1,:) <= 0)) error("bim2a_axisymmetric_advection_diffusion: the input mesh cannot intersect the rotation axis r=0."); endif nnodes = columns(mesh.p); nelem = columns(mesh.t); ## Turn scalar input to a vector of appropriate size if isscalar(alpha) alpha = alpha*ones(nelem,1); endif if isscalar(gamma) gamma = gamma*ones(nnodes,1); endif if isscalar(eta) eta = eta*ones(nnodes,1); endif if !( isvector(alpha) && isvector(gamma) && isvector(eta) ) error("bim2a_axisymmetric_advection_diffusion: coefficients are not valid vectors."); elseif length(alpha) != nelem error("bim2a_axisymmetric_advection_diffusion: length of alpha is not equal to the number of elements."); elseif length(gamma) != nnodes error("bim2a_axisymmetric_advection_diffusion: length of gamma is not equal to the number of nodes."); elseif length(eta) != nnodes error("bim2a_axisymmetric_advection_diffusion: length of eta is not equal to the number of nodes."); endif x = abs(mesh.p(1,:)); x = x(mesh.t(1:3,:)); y = mesh.p(2,:); y = y(mesh.t(1:3,:)); alphaareak = reshape (alpha.*mesh.area,1,1,nelem); shg = mesh.shg(:,:,:); ## Build local Laplacian matrix Lloc = zeros(3,3,nelem); for inode = 1:3 for jnode = 1:3 ginode(inode,jnode,:) = mesh.t(inode,:); gjnode(inode,jnode,:) = mesh.t(jnode,:); Lloc(inode,jnode,:) = sum( shg(:,inode,:) .* shg(:,jnode,:),1) .* alphaareak; endfor endfor if all(size(beta)==1) v12 = 0; v23 = 0; v31 = 0; elseif all(size(beta)==[2,nelem]) v12 = beta(1,:) .* (x(2,:)-x(1,:)) + beta(2,:) .* (y(2,:)-y(1,:)); v23 = beta(1,:) .* (x(3,:)-x(2,:)) + beta(2,:) .* (y(3,:)-y(2,:)); v31 = beta(1,:) .* (x(1,:)-x(3,:)) + beta(2,:) .* (y(1,:)-y(3,:)); elseif all(size(beta)==[nnodes,1]) betaloc = beta(mesh.t(1:3,:)); v12 = betaloc(2,:)-betaloc(1,:); v23 = betaloc(3,:)-betaloc(2,:); v31 = betaloc(1,:)-betaloc(3,:); else error("bim2a_axisymmetric_advection_diffusion: coefficient beta has wrong dimensions."); endif etaloc = eta(mesh.t(1:3,:)); eta12 = etaloc(2,:) - etaloc(1,:); eta23 = etaloc(3,:) - etaloc(2,:); eta31 = etaloc(1,:) - etaloc(3,:); r12 = (x(2,:) + x(1,:)) / 2; r23 = (x(3,:) + x(2,:)) / 2; r31 = (x(1,:) + x(3,:)) / 2; etalocm1 = bimu_logm(etaloc(2,:),etaloc(3,:)); etalocm2 = bimu_logm(etaloc(3,:),etaloc(1,:)); etalocm3 = bimu_logm(etaloc(1,:),etaloc(2,:)); gammaloc = gamma(mesh.t(1:3,:)); geloc = gammaloc.*etaloc; gelocm1 = bimu_logm (geloc(2,:), geloc(3,:)); gelocm2 = bimu_logm (geloc(3,:), geloc(1,:)); gelocm3 = bimu_logm (geloc(1,:), geloc(2,:)); [bp12,bm12] = bimu_bernoulli ((v12 - eta12) ./ etalocm3); [bp23,bm23] = bimu_bernoulli ((v23 - eta23) ./ etalocm1); [bp31,bm31] = bimu_bernoulli ((v31 - eta31) ./ etalocm2); bp12 = reshape(r12.*gelocm3.*etalocm3.*bp12,1,1,nelem).*Lloc(1,2,:); bm12 = reshape(r12.*gelocm3.*etalocm3.*bm12,1,1,nelem).*Lloc(1,2,:); bp23 = reshape(r23.*gelocm1.*etalocm1.*bp23,1,1,nelem).*Lloc(2,3,:); bm23 = reshape(r23.*gelocm1.*etalocm1.*bm23,1,1,nelem).*Lloc(2,3,:); bp31 = reshape(r31.*gelocm2.*etalocm2.*bp31,1,1,nelem).*Lloc(3,1,:); bm31 = reshape(r31.*gelocm2.*etalocm2.*bm31,1,1,nelem).*Lloc(3,1,:); Sloc(1,1,:) = (-bm12-bp31)./reshape(etaloc(1,:),1,1,nelem); Sloc(1,2,:) = bp12./reshape(etaloc(2,:),1,1,nelem); Sloc(1,3,:) = bm31./reshape(etaloc(3,:),1,1,nelem); Sloc(2,1,:) = bm12./reshape(etaloc(1,:),1,1,nelem); Sloc(2,2,:) = (-bp12-bm23)./reshape(etaloc(2,:),1,1,nelem); Sloc(2,3,:) = bp23./reshape(etaloc(3,:),1,1,nelem); Sloc(3,1,:) = bp31./reshape(etaloc(1,:),1,1,nelem); Sloc(3,2,:) = bm23./reshape(etaloc(2,:),1,1,nelem); Sloc(3,3,:) = (-bm31-bp23)./reshape(etaloc(3,:),1,1,nelem); A = sparse(ginode(:),gjnode(:),Sloc(:)); endfunction %!test %! n = 3; %! [mesh] = msh2m_structured_mesh(linspace(1,2,n+1),linspace(0,1,n+1),1,1:4); %! mesh = bim2c_mesh_properties(mesh); %! uex = @(r,z) exp(r); %! duexdr = @(r,z) uex(r,z); %! d2uexdr2 = @(r,z) uex(r,z); %! duexdz = @(r,z) 0*uex(r,z); %! d2uexdz2 = @(r,z) 0*uex(r,z); %! Dnodes = bim2c_unknowns_on_side(mesh,[2,4]); %! Nnodes = columns(mesh.p); %! Nelements = columns(mesh.t); %! Varnodes = setdiff(1:Nnodes,Dnodes); %! D = 1; vr = 1; vz = 0; %! alpha = D*ones(Nelements,1); %! gamma = ones(Nnodes,1); %! eta = ones(Nnodes,1); %! beta = 1/D*[vr*ones(1,Nelements); vz*ones(1,Nelements)]; %! f = @(r,z) -D./r.*duexdr(r,z) - D.*d2uexdr2(r,z) ... %! + vr./r .* uex(r,z) + vr * duexdr(r,z) ... %! - D.*d2uexdz2(r,z) + vz * duexdz(r,z); %! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh.p(1,:), mesh.p(2,:))); %! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta); %! u = zeros(Nnodes,1); u(Dnodes) = uex(mesh.p(1,Dnodes), mesh.p(2,Dnodes)); %! u(Varnodes) = S(Varnodes,Varnodes)\(rhs(Varnodes) - S(Varnodes,Dnodes)*u(Dnodes)); %! assert(u,uex(mesh.p(1,:), mesh.p(2,:))',1e-7) %!test %! n = 20; %! [mesh] = msh2m_structured_mesh(linspace(1,2,n+1),linspace(0,1,n+1),1,1:4); %! mesh = bim2c_mesh_properties(mesh); %! uex = @(r,z) exp(r) .* exp(1-z); %! duexdr = @(r,z) uex(r,z); %! d2uexdr2 = @(r,z) uex(r,z); %! duexdz = @(r,z) -uex(r,z); %! d2uexdz2 = @(r,z) uex(r,z); %! Dnodes = bim2c_unknowns_on_side(mesh,[1,2,3,4]); %! Nnodes = columns(mesh.p); %! Nelements = columns(mesh.t); %! Varnodes = setdiff(1:Nnodes,Dnodes); %! D = 1; vr = 1; vz = 1; %! alpha = D*ones(Nelements,1); %! gamma = ones(Nnodes,1); %! eta = ones(Nnodes,1); %! beta = 1/D*[vr*ones(1,Nelements); vz*ones(1,Nelements)]; %! f = @(r,z) -D./r.*duexdr(r,z) - D.*d2uexdr2(r,z) ... %! + vr./r .* uex(r,z) + vr * duexdr(r,z) ... %! - D.*d2uexdz2(r,z) + vz * duexdz(r,z); %! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh.p(1,:), mesh.p(2,:))); %! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta); %! u = zeros(Nnodes,1); u(Dnodes) = uex(mesh.p(1,Dnodes), mesh.p(2,Dnodes)); %! u(Varnodes) = S(Varnodes,Varnodes)\(rhs(Varnodes) - S(Varnodes,Dnodes)*u(Dnodes)); %! assert(u,uex(mesh.p(1,:), mesh.p(2,:))',1e-3) %!test %! n = 10; %! [mesh] = msh2m_structured_mesh(linspace(1,2,n+1),linspace(0,1,n+1),1,1:4); %! mesh = bim2c_mesh_properties(mesh); %! uex = @(r,z) exp(r) .* exp(1-z); %! duexdr = @(r,z) uex(r,z); %! d2uexdr2 = @(r,z) uex(r,z); %! duexdz = @(r,z) -uex(r,z); %! d2uexdz2 = @(r,z) uex(r,z); %! Dnodes = bim2c_unknowns_on_side(mesh,[1,2,3,4]); %! Nnodes = columns(mesh.p); %! Nelements = columns(mesh.t); %! Varnodes = setdiff(1:Nnodes,Dnodes); %! D = 1; %! alpha = D*ones(Nelements,1); %! gamma = ones(Nnodes,1); %! eta = ones(Nnodes,1); %! beta = 1/D * mesh.p(1,:)'; %! f = @(r,z) -D./r.*duexdr(r,z) - D.*d2uexdr2(r,z) ... %! + 1./r .* uex(r,z) + duexdr(r,z) ... %! - D.*d2uexdz2(r,z); %! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh.p(1,:), mesh.p(2,:))); %! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta); %! u = zeros(Nnodes,1); u(Dnodes) = uex(mesh.p(1,Dnodes), mesh.p(2,Dnodes)); %! u(Varnodes) = S(Varnodes,Varnodes)\(rhs(Varnodes) - S(Varnodes,Dnodes)*u(Dnodes)); %! assert(u,uex(mesh.p(1,:), mesh.p(2,:))',1e-3) %!test %! n = 10; %! [mesh] = msh2m_structured_mesh(linspace(1,2,n+1),linspace(0,1,n+1),1,1:4); %! mesh = bim2c_mesh_properties(mesh); %! uex = @(r,z) exp(r) .* exp(1-z); %! duexdr = @(r,z) uex(r,z); %! d2uexdr2 = @(r,z) uex(r,z); %! duexdz = @(r,z) -uex(r,z); %! d2uexdz2 = @(r,z) uex(r,z); %! Dnodes = bim2c_unknowns_on_side(mesh,[1,2,3,4]); %! Nnodes = columns(mesh.p); %! Nelements = columns(mesh.t); %! Varnodes = setdiff(1:Nnodes,Dnodes); %! D = 1; %! alpha = D*ones(Nelements,1); %! gamma = ones(Nnodes,1); %! eta = ones(Nnodes,1); %! beta = 1/D * 1/2*(mesh.p(1,:)').^2; %! f = @(r,z) 1./r.*(1+r).*(r-D) .* uex(r,z); %! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(mesh.p(1,:), mesh.p(2,:))); %! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta); %! u = zeros(Nnodes,1); u(Dnodes) = uex(mesh.p(1,Dnodes), mesh.p(2,Dnodes)); %! u(Varnodes) = S(Varnodes,Varnodes)\(rhs(Varnodes) - S(Varnodes,Dnodes)*u(Dnodes)); %! assert(u,uex(mesh.p(1,:), mesh.p(2,:))',1e-3) %!test %! n = 3; %! [mesh] = msh2m_structured_mesh(linspace(-2,-1,n+1),linspace(0,1,n+1),1,1:4); %! mesh = bim2c_mesh_properties(mesh); %! uex = @(r,z) exp(r); %! duexdr = @(r,z) uex(r,z); %! d2uexdr2 = @(r,z) uex(r,z); %! duexdz = @(r,z) 0*uex(r,z); %! d2uexdz2 = @(r,z) 0*uex(r,z); %! Dnodes = bim2c_unknowns_on_side(mesh,[2,4]); %! Nnodes = columns(mesh.p); %! Nelements = columns(mesh.t); %! Varnodes = setdiff(1:Nnodes,Dnodes); %! D = 1; vr = 1; vz = 0; %! alpha = D*ones(Nelements,1); %! gamma = ones(Nnodes,1); %! eta = ones(Nnodes,1); %! beta = 1/D*[vr*ones(1,Nelements); vz*ones(1,Nelements)]; %! f = @(r,z) -D./r.*duexdr(r,z) - D.*d2uexdr2(r,z) ... %! + vr./r .* uex(r,z) + vr * duexdr(r,z) ... %! - D.*d2uexdz2(r,z) + vz * duexdz(r,z); %! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(abs(mesh.p(1,:)), mesh.p(2,:))); %! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta); %! u = zeros(Nnodes,1); u(Dnodes) = uex(abs(mesh.p(1,Dnodes)), mesh.p(2,Dnodes)); %! u(Varnodes) = S(Varnodes,Varnodes)\(rhs(Varnodes) - S(Varnodes,Dnodes)*u(Dnodes)); %! assert(u,uex(abs(mesh.p(1,:)), mesh.p(2,:))',1e-7) %!test %! n = 10; %! [mesh] = msh2m_structured_mesh(linspace(-2,-1,n+1),linspace(0,1,n+1),1,1:4); %! mesh = bim2c_mesh_properties(mesh); %! uex = @(r,z) exp(r) .* exp(1-z); %! duexdr = @(r,z) uex(r,z); %! d2uexdr2 = @(r,z) uex(r,z); %! duexdz = @(r,z) -uex(r,z); %! d2uexdz2 = @(r,z) uex(r,z); %! Dnodes = bim2c_unknowns_on_side(mesh,[1,2,3,4]); %! Nnodes = columns(mesh.p); %! Nelements = columns(mesh.t); %! Varnodes = setdiff(1:Nnodes,Dnodes); %! D = 1; %! alpha = D*ones(Nelements,1); %! gamma = ones(Nnodes,1); %! eta = ones(Nnodes,1); %! beta = 1/D * 1/2*(mesh.p(1,:)').^2; %! f = @(r,z) 1./r.*(1+r).*(r-D) .* uex(r,z); %! rhs = bim2a_axisymmetric_rhs(mesh, ones(Nelements,1), f(abs(mesh.p(1,:)), mesh.p(2,:))); %! S = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta); %! u = zeros(Nnodes,1); u(Dnodes) = uex(abs(mesh.p(1,Dnodes)), mesh.p(2,Dnodes)); %! u(Varnodes) = S(Varnodes,Varnodes)\(rhs(Varnodes) - S(Varnodes,Dnodes)*u(Dnodes)); %! assert(u,uex(abs(mesh.p(1,:)), mesh.p(2,:))',1e-3) %!test %! [mesh] = msh2m_structured_mesh([0:.1:1],[0:.1:1],1,1:4); %! mesh = bim2c_mesh_properties(mesh); %! x = mesh.p(1,:)'; y = mesh.p(2,:)'; %! Dnodes = bim2c_unknowns_on_side(mesh,[1:4]); %! Nnodes = columns(mesh.p); Nelements = columns(mesh.t); %! alpha = ones(Nelements,1); eta=ones(Nnodes,1); %! beta = 0; %! gamma = ones(Nnodes,1); %! A = bim2a_axisymmetric_advection_diffusion(mesh,1,1,1,0); %! B = bim2a_axisymmetric_advection_diffusion(mesh,alpha,gamma,eta,beta); %! assert(A,B)