Bridging art and engineering using Escher-based virtual elements

The geometric shape of an element plays a key role in computational methods. Triangular and quadrilateral shaped elements are utilized by standard finite element methods. The pioneering work of Wachspress laid the foundation for polygonal interpolants which introduced polygonal elements. Tessellations may be considered as the next stage of element shape evolution. In this work, we investigate the topology optimization of tessellations as a means to coalesce art and engineering. We mainly focus on M.C. Escher’s tessellations using recognizable figures. To solve the state equation, we utilize a Mimetic Finite Difference inspired approach, known as the Virtual Element Method. In this approach, the stiffness matrix is constructed in such a way that the displacement patch test is passed exactly in order to ensure optimum numerical convergence rates. Prior to exploring the artistic aspects of topology optimization designs, numerical verification studies such as the displacement patch test and shear loaded cantilever beam bending problem are conducted to demonstrate the accuracy of the present approach in two-dimensions.