sbplib: +scheme/Elastic2dVariable.m comparison

comparison +scheme/Elastic2dVariable.m @ 813:b374a8aa9246 feature/grids

Correct interface penalty strength in Elastic2dVariable

author	Martin Almquist <malmquist@stanford.edu>
date	Wed, 08 Aug 2018 14:42:24 -0700
parents	1f6b2fb69225
children	386ef449df51 21394c78c72e

comparison

equal deleted inserted replaced

-:6b83dcb46f54
+:b374a8aa9246
 classdef Elastic2dVariable < scheme.Scheme
 % Discretizes the elastic wave equation:
 % rho u_{i,tt} = di lambda dj u_j + dj mu di u_j + dj mu dj u_i
 % opSet should be cell array of opSets, one per dimension. This
 % is useful if we have periodic BC in one direction.
 properties
 m % Number of points in each direction, possibly a vector
 % Traction operators used for BC
 T_l, T_r
 tau_l, tau_r
 H, Hi % Inner products
 phi % Borrowing constant for (d1 - e^T*D1) from R
 gamma % Borrowing constant for d1 from M
 H11 % First element of H
+% Borrowing from H, M, and R
+thH
+thM
+thR
 e_l, e_r
 d1_l, d1_r % Normal derivatives at the boundary
 E % E{i}^T picks out component i
 H_boundary % Boundary inner products
 % Kroneckered norms and coefficients
 RHOi_kron
 Hi_kron
 for i = 1:dim
 beta = ops{i}.borrowing.R.delta_D;
 obj.H11{i} = ops{i}.borrowing.H11;
 obj.phi{i} = beta/obj.H11{i};
 obj.gamma{i} = ops{i}.borrowing.M.d1;
+% Better names
+obj.thR{i} = ops{i}.borrowing.R.delta_D;
+obj.thM{i} = ops{i}.borrowing.M.d1;
+obj.thH{i} = ops{i}.borrowing.H11;
 end
 I = cell(dim,1);
 D1 = cell(dim,1);
 D2 = cell(dim,1);
 % Loop over components
 for i = 1:dim
 tau_l{j}{i} = sparse(m_tot,dim*m_tot);
 tau_r{j}{i} = sparse(m_tot,dim*m_tot);
 for k = 1:dim
 T_l{j}{i,k} = ...
 -d(i,j)*LAMBDA*(d(i,k)*e_l{k}*d1_l{k}' + db(i,k)*D1{k})...
 -d(j,k)*MU*(d(i,j)*e_l{i}*d1_l{i}' + db(i,j)*D1{i})...
 -d(i,k)*MU*e_l{j}*d1_l{j}';
 T_r{j}{i,k} = ...
 d(i,j)*LAMBDA*(d(i,k)*e_r{k}*d1_r{k}' + db(i,k)*D1{k})...
 +d(j,k)*MU*(d(i,j)*e_r{i}*d1_r{i}' + db(i,j)*D1{i})...
 +d(i,k)*MU*e_r{j}*d1_r{j}';
 tau_l{j}{i} = tau_l{j}{i} + T_l{j}{i,k}*E{k}';
 tau_r{j}{i} = tau_r{j}{i} + T_r{j}{i,k}*E{k}';
 end
 % Closure functions return the operators applied to the own domain to close the boundary
 % Penalty functions return the operators to force the solution. In the case of an interface it returns the operator applied to the other doamin.
 %       boundary            is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
 %       bc                  is a cell array of component and bc type, e.g. {1, 'd'} for Dirichlet condition
 %                           on the first component.
 %       data                is a function returning the data that should be applied at the boundary.
 %       neighbour_scheme    is an instance of Scheme that should be interfaced to.
 %       neighbour_boundary  is a string specifying which boundary to interface to.
 function [closure, penalty] = boundary_condition(obj, boundary, bc, tuning)
 default_arg('tuning', 1.2);
 d = @kroneckerDelta;  % Kronecker delta
 db = @(i,j) 1-d(i,j); % Logical not of Kronecker delta
 alpha = @(i,j) tuning*( d(i,j)* a_lambda*LAMBDA ...
 + d(i,j)* a_mu_i*MU ...
 + db(i,j)*a_mu_ij*MU );
 % Loop over components that Dirichlet penalties end up on
 for i = 1:dim
 C = T{k,i};
 A = -d(i,k)*alpha(i,j);
 B = A + C;
 closure = closure + E{i}*RHOi*Hi*B'*e*H_gamma*(e'*E{k}' );
 penalty = penalty - E{i}*RHOi*Hi*B'*e*H_gamma;
 end
 % Free boundary condition
 case {'F','f','Free','free','traction','Traction','t','T'}
 closure = closure - E{k}*RHOi*Hi*e*H_gamma* (e'*tau{k} );
 penalty = penalty + E{k}*RHOi*Hi*e*H_gamma;
 % Unknown boundary condition
 otherwise
 error('No such boundary condition: type = %s',type);
 % Operators and quantities that correspond to the own domain only
 Hi = obj.Hi;
 RHOi = obj.RHOi;
 dim = obj.dim;
 %--- Other operators ----
 m_tot_u = obj.grid.N();
 E = obj.E;
 LAMBDA_u = obj.LAMBDA;
 MU_u = obj.MU;
 LAMBDA_v = neighbour_scheme.LAMBDA;
 MU_v = neighbour_scheme.MU;
 lambda_v = e_v'*LAMBDA_v*e_v;
 mu_v = e_v'*MU_v*e_v;
 %-------------------------
 % Borrowing constants
-phi_u = obj.phi{j};
 h_u = obj.h(j);
-h11_u = obj.H11{j}*h_u;
+thR_u = obj.thR{j}*h_u;
-gamma_u = obj.gamma{j};
+thM_u = obj.thM{j}*h_u;
+thH_u = obj.thH{j}*h_u;
-phi_v = neighbour_scheme.phi{j_v};
 h_v = neighbour_scheme.h(j_v);
-h11_v = neighbour_scheme.H11{j_v}*h_v;
+thR_v = neighbour_scheme.thR{j_v}*h_v;
-gamma_v = neighbour_scheme.gamma{j_v};
+thH_v = neighbour_scheme.thH{j_v}*h_v;
+thM_v = neighbour_scheme.thM{j_v}*h_v;
-% E > sum_i 1/(2*alpha_ij)*(tau_i)^2
-function [alpha_ii, alpha_ij] = computeAlpha(phi,h,h11,gamma,lambda,mu)
+% alpha = penalty strength for normal component, beta for tangential
-th1 = h11/(2*dim);
+alpha_u = dim*lambda_u/(4*thH_u) + lambda_u/(4*thR_u) + mu_u/(2*thM_u);
-th2 = h11*phi/2;
+alpha_v = dim*lambda_v/(4*thH_v) + lambda_v/(4*thR_v) + mu_v/(2*thM_v);
-th3 = h*gamma;
+beta_u = mu_u/(2*thH_u) + mu_u/(4*thR_u);
-a1 = ( (th1 + th2)*th3*lambda + 4*th1*th2*mu ) / (2*th1*th2*th3);
+beta_v = mu_v/(2*thH_v) + mu_v/(4*thR_v);
-a2 = ( 16*(th1 + th2)*lambda*mu ) / (th1*th2*th3);
+alpha = alpha_u + alpha_v;
-alpha_ii = a1 + sqrt(a2 + a1^2);
+beta = beta_u + beta_v;
-alpha_ij = mu*(2/h11 + 1/(phi*h11));
-end
-[alpha_ii_u, alpha_ij_u] = computeAlpha(phi_u,h_u,h11_u,gamma_u,lambda_u,mu_u);
-[alpha_ii_v, alpha_ij_v] = computeAlpha(phi_v,h_v,h11_v,gamma_v,lambda_v,mu_v);
-sigma_ii = tuning*(alpha_ii_u + alpha_ii_v)/4;
-sigma_ij = tuning*(alpha_ij_u + alpha_ij_v)/4;
 d = @kroneckerDelta;  % Kronecker delta
 db = @(i,j) 1-d(i,j); % Logical not of Kronecker delta
-sigma = @(i,j) tuning*(d(i,j)*sigma_ii + db(i,j)*sigma_ij);
+strength = @(i,j) tuning*(d(i,j)*alpha + db(i,j)*beta);
 % Preallocate
 closure = sparse(dim*m_tot_u, dim*m_tot_u);
 penalty = sparse(dim*m_tot_u, dim*m_tot_v);
 % Loop over components that penalties end up on
 for i = 1:dim
-closure = closure - E{i}*RHOi*Hi*e*sigma(i,j)*H_gamma*e'*E{i}';
+closure = closure - E{i}*RHOi*Hi*e*strength(i,j)*H_gamma*e'*E{i}';
-penalty = penalty + E{i}*RHOi*Hi*e*sigma(i,j)*H_gamma*e_v'*E_v{i}';
+penalty = penalty + E{i}*RHOi*Hi*e*strength(i,j)*H_gamma*e_v'*E_v{i}';
 closure = closure - 1/2*E{i}*RHOi*Hi*e*H_gamma*e'*tau{i};
 penalty = penalty - 1/2*E{i}*RHOi*Hi*e*H_gamma*e_v'*tau_v{i};
 % Loop over components that we have interface conditions on
 for k = 1:dim
 closure = closure + 1/2*E{i}*RHOi*Hi*T{k,i}'*e*H_gamma*e'*E{k}';
 penalty = penalty - 1/2*E{i}*RHOi*Hi*T{k,i}'*e*H_gamma*e_v'*E_v{k}';
 end
 end
 end
 % Returns the coordinate number and outward normal component for the boundary specified by the string boundary.
 function [j, nj] = get_boundary_number(obj, boundary)
 switch boundary
 case {'w','W','west','West','s','S','south','South'}
 varargout{i} = obj.tau_l{j};
 case {'e', 'E', 'east', 'East','n', 'N', 'north', 'North'}
 varargout{i} = obj.tau_r{j};
 end
 otherwise
 error(['No such operator: operator = ' op{i}]);
 end
 end

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comparison +scheme/Elastic2dVariable.m @ 813:b374a8aa9246 feature/grids