Development of an IGA Based Topology Optimization Tool for Structural Problems with Geometric and Material Nonlinearities

Abstract

Isogeometric Analysis (IGA) is a recently developed tool capable of simulating Boundary Value Problems (BVP) effectively. Unlike Finite Element Method (FEM), IGA renders the derivatives of the field variables, e.g., strain, continuous across the knot spans, in addition to the primary field variables, e.g., displacement. This becomes possible because of the smoothness of NURBS used in the creation and analysis of the model. Moreover, FEM relies on approximate geometry for analysis; thus requiring larger number of elements and unknowns for complex geometries to achieve reasonable accuracy. Hence, an IGA based approach is convenient towards the design of complex shaped structures undergoing large deformation, that too in presence of nonlinear constitutive relation, e.g., Shape Memory Alloy (SMA). SMAs are popular for the large strain recovery capabilities, finding extensive applications in biomedical, aerospace, automobiles and robotics. Since most of these applications are of complex shape, particularly in biomedical devices, one requires an efficient mathematical model to analyze their thermomechanical behaviour. Moreover, an approach like Topology Optimization (TO) involves several repetitive analyses, and thus demand a very efficient model. In this work, an effort has been made to develop an IGA based TO tool for linear elastic as well as nonlinear material like SMA. Firstly, a Timoshenko beam formulation is implemented using IGA, following Total Lagrangian (TL) approach, where Green-Lagrange strain and second Piola-Kirchhoff stress measures are considered. Various elastic and SMA based problems are solved for validation purpose. The results are compared with that of the same obtained from the corotational formulation implemented using FEM. The results reveal the effectiveness of the IGA based tool in reducing the required number of degrees of freedom and thus the computational time.