P1-Nonconforming shell element and its application to topology optimization

The P1-nonconforming quadrilateral element developed by Park and Sheen [19] is employed for the formulation of a four-node degenerated shell element. The numerical stability of the P1-nonconforming quadrilateral element verified in plane elasticity problems and Stokes flow problems is investigated for the application of mitigating locking phenomena in shell problems. To facilitate the stiffness matrix computation for a non-flat general quadrilateral shell element, a nonparametric reference scheme using both affine transformation and bilinear transformation is adopted. Based on a field-consistency concept, the spurious constraints that cause locking are analyzed and an effective reduced integration scheme is found. The proposed shell element is applied to multi-physics topology optimization problems involving fluid analysis and shell analysis. For fluid analysis, the P1-nonconforming quadrilateral element is also adopted to utilize its volumetric locking-free property for incompressible materials. Vein layouts of leaves are designed by topology optimization and compared with natural vein layouts to verify the effectiveness of the proposed element.