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101 lines
3.0 KiB
101 lines
3.0 KiB
3 years ago
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% @Author: Jingqiao Hu
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% @Date: 2020-10-29 15:04:15
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% @LastEditTime: 2021-04-19 10:11:20
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%--------------------------
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% gradient of shape function on one ele, four nodes, N1 ~ N4
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%
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% gradN: [4*8]
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% [dN1dx, 0, dN2dx, 0, ...
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% 0, dN1dx, 0, dN2dx, ...
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% dN1dy, 0, dN2dy, 0, ...
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% 0, dN1dy, 0, dN2dy, ...]
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% This sequence is setted for computaion
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%
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% gradN2: [3*8]
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% [dN1dx, 0, dN2dx, 0, ...
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% 0, dN1dy, 0, dN2dy, ...
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% dN1dy, dN1dx, dN2dy, dN2dx]
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function [gradN, gradN2, N] = shape_gradient_ele(a, b, sample_num)
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if sample_num==1
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[gradN] = shape_gradient_1(a, b);
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N = [];
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gradN2 = [];
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else
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[gradN, gradN2, N] = shape_gradient_4(a, b);
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end
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end
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% 4*8
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function [gradN] = shape_gradient_1(a, b)
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GN_x = 0;
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GN_y = 0;
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dN_x = 1/4*[-(1-GN_x) (1-GN_x) (1+GN_x) -(1+GN_x)];
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dN_y = 1/4*[-(1-GN_y) -(1+GN_y) (1+GN_y) (1-GN_y)];
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J = [dN_x; dN_y]*[ -a a a -a;
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-b -b b b]';
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G = [inv(J) zeros(size(J));
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zeros(size(J)) inv(J)];
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dN(1,1:2:8) = dN_x; dN(3,1:2:8) = dN_y;
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dN(2,2:2:8) = dN_x; dN(4,2:2:8) = dN_y;
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B = G*dN;
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gradN = B;
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end
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% 4 * 8 & 3*8 for every cell
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function [gradN, gradN2, N] = shape_gradient_4(a, b)
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GaussNodes = [-1/sqrt(3); 1/sqrt(3)];
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gradN = cell(4,1);
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gradN2 = cell(4,1);
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L = [1 0 0 0; 0 0 0 1; 0 1 1 0];
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nodes = [-1, 1];
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N = cell(4,1);
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for i = 1:2
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for j = 1:2
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gp = 2*(i-1)+j;
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GN_x = GaussNodes(i);
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GN_y = GaussNodes(j);
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% GN_x = nodes(i); GN_y = nodes(j);
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dN_x = 1/4*[-(1-GN_x) (1-GN_x) (1+GN_x) -(1+GN_x)];
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dN_y = 1/4*[-(1-GN_y) -(1+GN_y) (1+GN_y) (1-GN_y)];
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J = [dN_x; dN_y]*[ -a a a -a;
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-b -b b b]';
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G = [inv(J) zeros(size(J));
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zeros(size(J)) inv(J)];
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dN1(1,1:2:8) = dN_x; dN1(2,1:2:8) = dN_y;
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dN1(3,2:2:8) = dN_x; dN1(4,2:2:8) = dN_y;
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% dNdx = 4*8
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B = G*dN1;
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gradN{gp} = B([1,3,2,4],:);
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% dNdx = 3*8
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dN1(1,1:2:8) = dN_x; dN1(2,1:2:8) = dN_y;
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dN1(3,2:2:8) = dN_x; dN1(4,2:2:8) = dN_y;
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B2 = L*G*dN1;
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gradN2{gp} = B2;
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% N = 2*8
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GN_x = nodes(i); GN_y = nodes(j);
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N1 = (1-GN_x)*(1-GN_y)/4;
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N2 = (1+GN_x)*(1-GN_y)/4;
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N3 = (1+GN_x)*(1+GN_y)/4;
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N4 = (1-GN_x)*(1+GN_y)/4;
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N{gp} = [N1, 0, N2, 0, N3, 0, N4, 0;
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0, N1, 0, N2, 0, N3, 0, N4];
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% N2{gp} = [N1, N1;
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% N2, N2;
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% N3, N3;
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% N4, N4;]; % 4*2
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end
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end
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end
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