| 1 | function md=diagnostic(md)
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| 2 | %DIAGNOSTIC - compute the velocity field of a 2d model using MacAyeal solution
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| 3 | %
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| 4 | % this routine solves the problem using MacAyeal's model. It calculates the velocity
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| 5 | % field corresponding to the parameters and the geometry given by the model md
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| 6 | %
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| 7 | % Usage:
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| 8 | % md=diagnostic(md)
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| 9 |
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| 10 | %First check we do have the correct argument number
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| 11 | if ((nargin~=1) || (nargout~=1)),
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| 12 | macayealdiagnosticusage();
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| 13 | error('macayealdiagnostic error message: incorrect number of input and output arguments');
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| 14 | end
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| 15 |
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| 16 | if ~isa(md,'model'),
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| 17 | macayealdiagnosticusage();
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| 18 | error('macayealdiagnostic error message: input argument is not a @model object');
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| 19 | end
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| 20 |
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| 21 | %start timing
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| 22 | t1=clock;
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| 23 |
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| 24 | %Transfer model fields into matlab variables
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| 25 | x=md.x;
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| 26 | y=md.y;
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| 27 |
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| 28 | index=md.elements;index=sort(index,2); %necessary
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| 29 | nods=md.numberofgrids;
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| 30 | nel=md.numberofelements;
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| 31 | z_thick=md.thickness;
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| 32 | z_surf=md.surface;
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| 33 | z_bed=md.bed;
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| 34 | z_thick_bar=(z_thick(index(:,1))+z_thick(index(:,2))+z_thick(index(:,3)))/3;
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| 35 | rho_ice=md.rho_ice;
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| 36 | rho_water=md.rho_water;
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| 37 | g=md.g;
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| 38 | viscosity_overshoot=md.viscosity_overshoot;
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| 39 | index_icefront=md.segmentonneumann_diag; index_icefront=index_icefront(:,1:2); %we strip the last column, which holds the element number for the boundary segment
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| 40 | nodes_on_boundary=md.gridonboundary;
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| 41 | nodes_on_icefront=zeros(nods,1); nodes_on_icefront(index_icefront)=1;
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| 42 | node_on_dirichlet=md.gridondirichlet_diag;
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| 43 | nodes_on_icesheet=md.gridonicesheet;
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| 44 | element_on_icesheet=md.elementonicesheet;
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| 45 | qcoeff=md.q;
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| 46 | pcoeff=md.p;
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| 47 | drag_type=md.drag_type;
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| 48 | drag=md.drag;
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| 49 |
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| 50 | criterion_rel=md.eps_rel;
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| 51 | criterion_abs=md.eps_abs;
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| 52 | yts=md.yts;
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| 53 | B=md.B; B_bar=(B(index(:,1))+B(index(:,2))+B(index(:,3)))/3;
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| 54 | glen_coeff=md.n;
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| 55 |
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| 56 | %average of p and q over the grids (size nel->nods)
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| 57 | pcoeff_grid=zeros(nods,1);
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| 58 | qcoeff_grid=zeros(nods,1);
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| 59 | for i=1:nods
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| 60 | %1: find the elements that contain the grid i
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| 61 | neighbors_gridi=[];
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| 62 | for j=1:3
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| 63 | neighbors_gridi=[neighbors_gridi find(index(:,j)==i)'];
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| 64 | end
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| 65 | numberofneighbors_gridi=length(neighbors_gridi);
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| 66 | %2 retrieve the value of p and q over each of these elements. The average is
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| 67 | %plugged into dbx_grid
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| 68 | qcoeff_grid(i)=sum(qcoeff(neighbors_gridi))/numberofneighbors_gridi;
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| 69 | pcoeff_grid(i)=sum(pcoeff(neighbors_gridi))/numberofneighbors_gridi;
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| 70 | end
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| 71 |
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| 72 | %Build length_icefront and normal_icefront:
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| 73 | [length_icefront,normal_icefront]=buildicefrontnormal(x,y,index_icefront);
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| 74 |
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| 75 | %Start building areas
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| 76 | aire=zeros(md.numberofelements,1);
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| 77 |
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| 78 | for n=1:nel
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| 79 | aire(n)=1/2 * det([1 1 1;x(index(n,:))';y(index(n,:))']);
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| 80 | end
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| 81 |
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| 82 | aire=abs(aire); % if index is sorted from its original value, then aire could be negative
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| 83 |
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| 84 | alpha=zeros(nel,3);
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| 85 | beta=zeros(nel,3);
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| 86 | gamma=zeros(nel,3);
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| 87 |
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| 88 | for n=1:nel
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| 89 | X=inv([x(index(n,:)) y(index(n,:)) ones(3,1)]);
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| 90 | alpha(n,:)=X(1,:);
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| 91 | beta(n,:)=X(2,:);
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| 92 | gamma(n,:)=X(3,:);
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| 93 | end
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| 94 |
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| 95 | clear X
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| 96 |
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| 97 | %Initialize viscosity
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| 98 | nu_bar=viscosity(index,nel,alpha,beta,{},{},B_bar,glen_coeff);
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| 99 |
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| 100 |
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| 101 | % Do once and for all the initial computation of matrix-locations:
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| 102 |
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| 103 | row_location=zeros(nel*3*3,1);
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| 104 | col_location=zeros(nel*3*3,1);
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| 105 | row_location_AD=zeros(nel*6,1);
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| 106 | col_location_AD=zeros(nel*6,1);
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| 107 |
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| 108 | count=-nel+1;
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| 109 |
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| 110 | for i=1:3
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| 111 | for j=1:3
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| 112 | count=count+nel;
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| 113 | row_location(count:count+nel-1)=index(:,i);
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| 114 | col_location(count:count+nel-1)=index(:,j);
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| 115 | end
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| 116 | end
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| 117 |
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| 118 |
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| 119 | count=-nel+1;
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| 120 |
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| 121 | for i=1:3
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| 122 | for j=i:3
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| 123 | count=count+nel;
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| 124 | row_location_AD(count:count+nel-1)=index(:,i);
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| 125 | col_location_AD(count:count+nel-1)=index(:,j);
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| 126 | end
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| 127 | end
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| 128 |
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| 129 |
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| 130 |
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| 131 | permanent_pieces_of_A=zeros(nel*6,1);
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| 132 | permanent_pieces_of_B=zeros(nel*3*3,1);
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| 133 | permanent_pieces_of_C=zeros(nel*3*3,1);
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| 134 | permanent_pieces_of_D=zeros(nel*6,1);
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| 135 |
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| 136 | count=-nel+1;
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| 137 |
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| 138 | for i=1:3
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| 139 | for j=i:3
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| 140 | count=count+nel;
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| 141 | permanent_pieces_of_A(count:count+nel-1)= z_thick_bar .* aire ...
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| 142 | .*(2*alpha(:,i).*alpha(:,j) + 1/2*beta(:,i).*beta(:,j));
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| 143 | % This loop structure works only when index is sorted
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| 144 | permanent_pieces_of_D(count:count+nel-1)= z_thick_bar .* aire ...
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| 145 | .*(2*beta(:,i).*beta(:,j) + 1/2*alpha(:,i).*alpha(:,j));
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| 146 |
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| 147 | end
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| 148 | end
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| 149 |
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| 150 | count=-nel+1;
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| 151 |
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| 152 | for i=1:3
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| 153 | for j=1:3
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| 154 | count=count+nel;
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| 155 |
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| 156 | permanent_pieces_of_B(count:count+nel-1)= z_thick_bar .* aire ...
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| 157 | .*(alpha(:,j).*beta(:,i) + 1/2*beta(:,j).*alpha(:,i));
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| 158 |
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| 159 | permanent_pieces_of_C(count:count+nel-1)= z_thick_bar .* aire ...
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| 160 | .*(beta(:,j).*alpha(:,i) + 1/2*alpha(:,j).*beta(:,i));
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| 161 |
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| 162 |
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| 163 | end
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| 164 | end
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| 165 |
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| 166 | % Step 3 -- Set up right-hand side of the problem.
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| 167 |
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| 168 | % (Note to myself: to avoid vector dependency problem, yet still vectorize,
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| 169 | % I treat the Rhs vector as a sparse matrix
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| 170 |
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| 171 | Rhs_x=zeros(nel*27,1);
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| 172 | Rhs_y=zeros(nel*27,1);
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| 173 | Rhs_y=ones(nel*27,1);
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| 174 | row_rhs=zeros(nel*27,1);
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| 175 |
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| 176 |
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| 177 | count=-nel+1;
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| 178 | for i=1:3
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| 179 | for n=1:3
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| 180 | for m=1:3
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| 181 | count=count+nel;
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| 182 |
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| 183 | Rhs_x(count:count+nel-1)= -rho_ice * g * ...
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| 184 | z_thick(index(:,n)).*z_surf(index(:,m)) ...
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| 185 | .* aire(:) .* alpha(:,m) * ( (n==i)/6 + (n~=i)/12 );
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| 186 |
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| 187 | Rhs_y(count:count+nel-1)= -rho_ice * g * ...
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| 188 | z_thick(index(:,n)).*z_surf(index(:,m)) ...
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| 189 | .* aire(:) .* beta(:,m) .* ( (n==i)/6 + (n~=i)/12 );
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| 190 |
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| 191 | row_rhs(count:count+nel-1)=index(:,i);
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| 192 |
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| 193 | end
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| 194 | end
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| 195 | end
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| 196 |
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| 197 | Rhs=full([sparse(row_rhs,ones(nel*27,1),Rhs_x,nods,1)
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| 198 | sparse(row_rhs,ones(nel*27,1),Rhs_y,nods,1)]);
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| 199 |
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| 200 | for k=1:2
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| 201 | for l=1:2
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| 202 | for j=1:2
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| 203 | Rhs(index_icefront(:,k))=Rhs(index_icefront(:,k)) + ...
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| 204 | (rho_ice*g/2*z_thick(index_icefront(:,l)).*z_thick(index_icefront(:,j)) ...
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| 205 | + rho_water*g/2*(md.gridoniceshelf(index_icefront(:,k))) ...
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| 206 | .*(-min(0,z_bed(index_icefront(:,l))).*min(0,z_bed(index_icefront(:,j))) ...
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| 207 | +min(0,z_thick(index_icefront(:,l))+z_bed(index_icefront(:,l)))...
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| 208 | .*min(0,z_thick(index_icefront(:,j))+z_bed(index_icefront(:,j)))))...
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| 209 | .*normal_icefront(:,1).*length_icefront(:) ...
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| 210 | /(4*(l==k & j==k) + 12*(l~=k | j~=k) );
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| 211 |
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| 212 | Rhs(index_icefront(:,k)+nods)=Rhs(index_icefront(:,k)+nods) + ...
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| 213 | (rho_ice*g/2*z_thick(index_icefront(:,l)).*z_thick(index_icefront(:,j)) ...
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| 214 | + rho_water*g/2*(md.gridoniceshelf(index_icefront(:,k))) ...
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| 215 | .*(-min(0,z_bed(index_icefront(:,l))).*min(0,z_bed(index_icefront(:,j))) ...
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| 216 | +min(0,z_thick(index_icefront(:,l))+z_bed(index_icefront(:,l)))...
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| 217 | .*min(0,z_thick(index_icefront(:,j))+z_bed(index_icefront(:,j)))))...
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| 218 | .*normal_icefront(:,2).*length_icefront(:) ...
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| 219 | /(4*(l==k & j==k) + 12*(l~=k | j~=k) );
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| 220 |
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| 221 | end
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| 222 | end
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| 223 | end
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| 224 |
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| 225 | clear Rhs_x Rhs_y row_rhs
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| 226 |
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| 227 | % Step 4 -- Create the parsing matrix to "wring out"
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| 228 | % the known boundary conditions
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| 229 |
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| 230 | % kinematic condition (u,v=0) specified at non-ice-front boundaries;
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| 231 | num_specified= 2*sum(node_on_dirichlet);
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| 232 | row=zeros(2*nods-num_specified,1);
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| 233 | col=zeros(2*nods-num_specified,1);
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| 234 | value=zeros(2*nods-num_specified,1);
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| 235 | count=0;
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| 236 | for n=1:nods
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| 237 | if(~node_on_dirichlet(n))
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| 238 | count=count+1;
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| 239 | row(count)=count;
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| 240 | col(count)=n;
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| 241 | value(count)=1;
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| 242 | end
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| 243 | end
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| 244 | for n=1:nods
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| 245 | if (~node_on_dirichlet(n))
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| 246 | count=count+1;
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| 247 | row(count)=count;
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| 248 | col(count)=nods+n;
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| 249 | value(count)=1;
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| 250 | end
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| 251 | end
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| 252 |
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| 253 | P=sparse(row,col,value,2*nods-num_specified,2*nods);
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| 254 |
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| 255 |
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| 256 | specified_velocity=zeros(2*nods,1);
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| 257 | pos=find(node_on_dirichlet);
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| 258 | specified_velocity(pos)=md.dirichletvalues_diag(pos,1)/md.yts;
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| 259 | specified_velocity(nods+pos)=md.dirichletvalues_diag(pos,2)/md.yts;
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| 260 |
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| 261 |
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| 262 | % ######## an attention grabbing break ########
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| 263 | % Iterate on flow law till converged
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| 264 |
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| 265 | converged_yet=0;
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| 266 | loop=0;
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| 267 |
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| 268 | u_old=zeros(nods,1);
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| 269 | v_old=zeros(nods,1);
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| 270 | convergence_count=1;
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| 271 |
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| 272 | while (~converged_yet)
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| 273 |
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| 274 | if (loop>(100))
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| 275 | warning('Maximum Viscosity Iterations Reached.')
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| 276 | break;
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| 277 | end;
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| 278 | loop=loop+1;
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| 279 |
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| 280 | nu_bar_oldvalue=nu_bar;
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| 281 |
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| 282 | % Step 4 -- Set up stress-balance matrix (whos solution is the velocity
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| 283 | % field):
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| 284 |
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| 285 | matrix_value_A=zeros(nel*6,1);
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| 286 | matrix_value_B=zeros(nel*3*3,1);
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| 287 | matrix_value_C=zeros(nel*3*3,1);
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| 288 | matrix_value_D=zeros(nel*6,1);
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| 289 |
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| 290 | count=-nel+1;
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| 291 |
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| 292 | for i=1:3
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| 293 | for j=i:3
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| 294 | count=count+nel;
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| 295 |
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| 296 | matrix_value_A(count:count+nel-1)=nu_bar .* ...
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| 297 | permanent_pieces_of_A(count:count+nel-1);
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| 298 |
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| 299 | matrix_value_D(count:count+nel-1)=nu_bar .* ...
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| 300 | permanent_pieces_of_D(count:count+nel-1);
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| 301 |
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| 302 |
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| 303 | end
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| 304 | end
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| 305 |
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| 306 | count=-nel+1;
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| 307 |
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| 308 | for i=1:3
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| 309 | for j=1:3
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| 310 | count=count+nel;
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| 311 |
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| 312 | matrix_value_B(count:count+nel-1)=nu_bar .* ...
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| 313 | permanent_pieces_of_B(count:count+nel-1);
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| 314 |
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| 315 | matrix_value_C(count:count+nel-1)=nu_bar .* ...
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| 316 | permanent_pieces_of_C(count:count+nel-1);
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| 317 |
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| 318 |
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| 319 | end
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| 320 | end
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| 321 |
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| 322 |
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| 323 |
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| 324 | A=sparse(row_location_AD,col_location_AD,matrix_value_A,nods,nods);
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| 325 | A=A+triu(A,1)';
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| 326 | B=sparse(row_location,col_location,matrix_value_B,nods,nods);
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| 327 | C=sparse(row_location,col_location,matrix_value_C,nods,nods);
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| 328 | D=sparse(row_location_AD,col_location_AD,matrix_value_D,nods,nods);
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| 329 | D=D+triu(D,1)';
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| 330 |
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| 331 |
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| 332 | F=[A C
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| 333 | B D];
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| 334 |
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| 335 | %Now, take care of the basal friction if there is any: to make things easier, we translate u = k *Neff^(-q)* sigma^p into
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| 336 | % sigma= drag^2 * Neff ^(r) * u^s with r=q/p and s=1/p : */
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| 337 |
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| 338 | if (drag_type==2),
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| 339 | %compute coeffs:
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| 340 | rcoeff=qcoeff_grid./pcoeff_grid;
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| 341 | scoeff=1./pcoeff_grid;
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| 342 |
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| 343 | %initialization of basal drag stiffness
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| 344 | Dragoperator=spalloc(2*nods,2*nods,0);
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| 345 |
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| 346 | if loop~=1,
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| 347 |
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| 348 | %retrieve the velocity magnitude
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| 349 | velocity_mag=sqrt(solution(1:nods).^2+solution(nods+1:2*nods).^2);
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| 350 |
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| 351 | %Computation of the effective pressure
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| 352 | Neff=g*(rho_ice*z_thick+rho_water*z_bed);
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| 353 |
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| 354 | %If effective pressure becomes negative, sliding becomes unstable (Paterson 4th edition p 148). This is because
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| 355 | %the water pressure is so high, the ice sheet elevates over its ice bumps and slides. But the limit behaviour
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| 356 | %for this should be an ice shelf sliding (no basal drag). Therefore, for any effective pressure Neff < 0, we should
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| 357 | %replace it by Neff=0 (ie, equival it to an ice shelf)*/
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| 358 | pos=find(Neff<0);
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| 359 | Neff(pos)=0;
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| 360 |
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| 361 | %Basal drag coefficient: Tau_x=-alpha^2 u, Tau_y=-alpha^2 v (See
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| 362 | %MacAyeal)
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| 363 | alpha2=(drag.^2).*(Neff.^rcoeff).*(velocity_mag.^(scoeff-1));
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| 364 |
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| 365 | %stiffness due to basal drag
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| 366 | %initialization
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| 367 | count=0;
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| 368 | value=zeros(nel,27);
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| 369 | row=zeros(nel,27);
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| 370 | col=zeros(nel,27);
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| 371 |
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| 372 | for m=1:3
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| 373 | for k=1:3
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| 374 | for l=1:3
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| 375 | if ( (m==k) + (m==l) + (l==k) )==3
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| 376 | fac=1/10;
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| 377 | elseif ( (m==k) + (m==l) + (l==k) )==1
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| 378 | fac=1/30;
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| 379 | else
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| 380 | fac=1/60;
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| 381 | end
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| 382 |
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| 383 | count=count+1;
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| 384 | row(:,count)=index(:,m);
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| 385 | col(:,count)=index(:,k);
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| 386 | value(:,count)=fac*aire(:).* ...
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| 387 | (alpha2(index(:,l)));%.*element_on_icesheet(index(:,l));
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| 388 |
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| 389 | end
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| 390 | end
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| 391 | end
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| 392 |
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| 393 | Dragoperator=[sparse(row,col,value,nods,nods) spalloc(nods,nods,0)
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| 394 | spalloc(nods,nods,0) sparse(row,col,value,nods,nods)];
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| 395 | end
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| 396 |
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| 397 | F=F+Dragoperator; %plug into the global stiffness matrix
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| 398 |
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| 399 | end
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| 400 |
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| 401 |
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| 402 | Rhs_parsed=P*(Rhs - F*specified_velocity);
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| 403 | F=P*F*P';
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| 404 |
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| 405 |
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| 406 | % Step 5 -- Solve the problem:
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| 407 |
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| 408 | % Digression: if need be clear up some memory:
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| 409 |
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| 410 | clear matrix_value_A ...
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| 411 | matrix_value_B matrix_value_C matrix_value_D A B C D
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| 412 |
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| 413 | % We can use either the LU or the Cholesky decomposition, but the
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| 414 | % Cholesky decomposition is twice as efficient as LU for symmetric
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| 415 | % definite positive matrix
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| 416 |
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| 417 | if strcmpi(md.solver_type,'lu'),
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| 418 | % Solve by LU decomposition.
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| 419 | [L,U] = lu(F);
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| 420 | solution= U\(L\Rhs_parsed);
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| 421 | elseif strcmpi(md.solver_type,'cholesky'),
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| 422 | % Solve by Choleski decomposition.
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|---|
| 423 | L = chol(F);
|
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| 424 | solution= L\(L'\Rhs_parsed);
|
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| 425 | else
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| 426 | % use matlab's generic solver
|
|---|
| 427 | solution = F\Rhs_parsed;
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|---|
| 428 | end
|
|---|
| 429 |
|
|---|
| 430 |
|
|---|
| 431 |
|
|---|
| 432 | %Add spcs to the calculated solution
|
|---|
| 433 | solution=P'*solution + specified_velocity;
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|---|
| 434 |
|
|---|
| 435 | %Recover solution vector
|
|---|
| 436 | u=solution(1:nods);
|
|---|
| 437 | v=solution(nods+1:2*nods);
|
|---|
| 438 |
|
|---|
| 439 | %Compute viscosity
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|---|
| 440 | nu_bar=viscosity(index,nel,alpha,beta,u,v,B_bar,glen_coeff);
|
|---|
| 441 | change=1 - nu_bar_oldvalue./nu_bar;
|
|---|
| 442 |
|
|---|
| 443 | %Update viscosity
|
|---|
| 444 | nu_bar=nu_bar.*(1+viscosity_overshoot*change);
|
|---|
| 445 | location=find(nu_bar<=0);
|
|---|
| 446 | nu_bar(location)=-nu_bar(location);
|
|---|
| 447 |
|
|---|
| 448 | %Test for direct shooting convergence
|
|---|
| 449 | if convergence_count>1,
|
|---|
| 450 |
|
|---|
| 451 | ug=[u_old;v_old];
|
|---|
| 452 | nug=norm(ug,2);
|
|---|
| 453 | dug=[u; v]-[u_old; v_old];
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|---|
| 454 | ndug=norm(dug,2);
|
|---|
| 455 | relative_change=ndug/nug;
|
|---|
| 456 |
|
|---|
| 457 |
|
|---|
| 458 | %Figure out if viscosity converged
|
|---|
| 459 | if relative_change<criterion_rel,
|
|---|
| 460 | disp(sprintf('%s %g %s %g %s',' Convergence criterion: norm(du)/norm(u)=',relative_change,' < ',criterion_rel,' m/yr'));
|
|---|
| 461 | converged_yet=1;
|
|---|
| 462 | else
|
|---|
| 463 | disp(sprintf('%s %g %s %g %s',' Convergence criterion: norm(du)/norm(u)=',relative_change,' > ',criterion_rel,' m/yr'));
|
|---|
| 464 | converged_yet=0;
|
|---|
| 465 | end
|
|---|
| 466 |
|
|---|
| 467 | if ~isnan(criterion_abs),
|
|---|
| 468 | change=max(abs(dug))*yts;
|
|---|
| 469 | if change<criterion_abs
|
|---|
| 470 | disp(sprintf('%s %g %s %g %s',' Convergence criterion: max(du)=',change,' < ',criterion_abs,' m/yr'));
|
|---|
| 471 | else
|
|---|
| 472 | disp(sprintf('%s %g %s %g %s',' Convergence criterion: max(du)=',change,' > ',criterion_abs,' m/yr'));
|
|---|
| 473 | converged_yet=0;
|
|---|
| 474 | end
|
|---|
| 475 | end
|
|---|
| 476 | else
|
|---|
| 477 | converged_yet=0;
|
|---|
| 478 | end
|
|---|
| 479 |
|
|---|
| 480 | u_old=u;
|
|---|
| 481 | v_old=v;
|
|---|
| 482 |
|
|---|
| 483 | convergence_count=convergence_count+1;
|
|---|
| 484 |
|
|---|
| 485 | end % This end statement terminates the "while" command way above
|
|---|
| 486 |
|
|---|
| 487 | %Load results onto md:
|
|---|
| 488 | md.vx=u*yts;
|
|---|
| 489 | md.vy=v*yts;
|
|---|
| 490 | md.vel=sqrt(md.vx.^2+md.vy.^2);
|
|---|
| 491 |
|
|---|
| 492 | %stop timing
|
|---|
| 493 | t2=clock;
|
|---|
| 494 |
|
|---|
| 495 | disp(sprintf('\n%s\n',['solution converged in ' num2str(etime(t2,t1)) ' seconds']));
|
|---|
| 496 |
|
|---|
| 497 | function macayealdiagnosticusage();
|
|---|
| 498 | disp('md=macayealdiagnostic(md)');
|
|---|
| 499 | disp(' where md is a structure of class @model');
|
|---|
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