Evolutionary topology optimization of continuum structures using X-FEM and isovalues of structural performance

In the last three decades, advances in modern manufacturing processes, such as additive manufacturing (AM) on one hand and computational power on the other hand, has resulted in a surge of interest in topology optimization as a means of designing high performance components with high degrees of geometrical complexity. Topology optimization seeks to find the best design for a structure by optimally distributing material in a design space. Therefore not only the shape and size of the structure, but also the connectivity of the structure changes during the topology optimization process. As a result, the solution of a topology optimization problem might be represented with a high degree of geometrical complexity as it is not dependent on the initial geometry. The finite element method (FEM) is a powerful numerical analysis technique that was developed to solve complex solid mechanics problems. Many topology optimization approaches use FEM to calculate the response of the structure during the optimization process and some of them, called “element based-methods”, are integrated with FEM to use the properties of finite elements as design variables in the optimization. The solutions of such approaches are usually represented by a uniform finite element mesh that bears no relation to the final geometry and hence they don’t provide an accurate representation of the design boundary. The solution from topology optimization must therefore go through further post processing stages to obtain a manufacturable design. The post processing stages which can include smoothing and shape optimization are costly and time-consuming and may result in the structure becoming less optimal. With traditional manufacturing processes this is acceptable as the manufacturing constraints prevent the optimized design from being manufactured so some re-analysis is necessary. With additive manufacturing, however, this restriction is removed, which means a topology optimization resulting in a manufacturable design is highly desirable. Evolutionary structural optimization (ESO) is an element based topology optimization approach which operates by systematically removing inefficient material from the structure until the optimization objective achieves convergence. Due to the intuitive nature of ESO, this method is simple to be programed and can be easily integrated with FEM or other numerical analysis techniques; thus it is suitable for complex geometries represented with FEM. During the last two decades ESO and its extensions, such as bi-directional ESO (BESO), have been successfully used for many topology optimization problems such as stiffness design, design of compliant mechanisms, heat conduction problems and frequency problems. However, being an element based method, the drawback of poor boundary representation remains. The extended finite element method (X-FEM) is an extension of the classical FEM that was developed to represent discontinuities, such as cracks and material-void interfaces, inside finite elements. X-FEM can be employed in topology optimization problems to handle the material-void discontinuity introduced by the evolving boundary during the optimization process which potentially enables a sub-element boundary representation. This requires an implicit boundary representation, such as level-set method with the benefits of better computational accuracy through the optimization, more optimized solution and smoother boundaries for direct to manufacture. In this work a new method of evolutionary structural optimization is proposed in which X-FEM is employed for the more smooth and accurate representation of the design boundary. Linear finite elements are used to discretize the design space. These include 4-node quadrilateral elements in 2D modelling and 8-node hexahedral elements in 3D modelling. To implement the X-FEM, an implicit boundary representation using isoline and isosurface approaches is used. The proposed method which is called “Iso-XFEM” is implemented for various topology optimization problems, including the stiffness design of 2D and 3D structures, stiffness design with additional displacement constraint and topology optimization of geometrically nonlinear problems. The solutions of the Iso-XFEM method are compared with those obtained using BESO, as a representative FE based method. The results confirm a significant improvement in boundary representation of the solutions when compared against BESO, and also demonstrate the feasibility of the application of the proposed method to complex real-life structures and to different objectives. All the programs used to generate topology optimised solutions using the proposed method and its modifications are developed by the author. These include topology optimization codes, linear and non-linear FEA, and 2D and 3D X-FEM integration schemes