# If only the first derivatives of ux and uy appear in the weak forms, then their interpolation must be at least bilinear.

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If only the first derivatives of ux and uy appear in the weak forms, then their interpolation must be at least bilinear.

(a) True

(b) False

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This interesting question is from Plane Elasticity topic in chapter Plane Elasticity of Finite Element Method

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Right choice is (a) True

Best explanation: An examination of the weak form, 0=∫Ωehe$[\frac{\partial w_1}{\partial x}(c_{11}\frac{\partial u_x}{\partial x}+c_{12}\frac{\partial u_y}{\partial y}) + c_{66}\frac{\partial w_1}{\partial y} (\frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x})$+ρw1ux]dxdy-∫Ωehew1fxdxdy-∮┌dhew1txds reveals that only first derivatives of ux and uy with respect to x and y appear respectively. Therefore, ux and uy must be approximated by the Lagrange family of interpolation functions, and at least a bilinear (i.e., linear both in x and y) interpolation is required.

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