Isogeometric topology optimization for continuum structures using density distribution function

Abstract

Summary This paper will propose a more effective and efficient topology optimization method based on isogeometric analysis, termed as isogeometric topology optimization (ITO), for continuum structures using an enhanced density distribution function (DDF). The construction of the DDF involves two steps. (1) Smoothness: the Shepard function is firstly utilized to improve the overall smoothness of nodal densities. Each nodal density is assigned to a control point of the geometry. (2) Continuity: the high‐order NURBS basis functions are linearly combined with the smoothed nodal densities to construct the DDF for the design domain. The nonnegativity, partition of unity, and restricted bounds [0, 1] of both the Shepard function and NURBS basis functions can guarantee the physical meaning of material densities in the design. A topology optimization formulation to minimize the structural mean compliance is developed based on the DDF and isogeometric analysis to solve structural responses. An integration of the geometry parameterization and numerical analysis can offer the unique benefits for the optimization. Several 2D and 3D numerical examples are performed to demonstrate the effectiveness and efficiency of the proposed ITO method, and the optimized 3D designs are prototyped using the Selective Laser Sintering technique.

Nodal Cosine Sine Material Interpolation in Multi Objective Topology Optimization With the Global Criteria Method for Linear Elasto Static, Heat Transfer, Potential Flow and Binary Cross Entropy Sharpening

“…interpolation with meshless field nodes and computational nodes. In a more recent article, topology optimization is applied using isogeometric analysis (IGA) (Dedè et al, 2012;Gao et al, 2020Gao et al, , 2019Hassani et al, 2013;Liu et al, 2018;Seo et al, 2010). The isogeometric analysis uses basis functions such as NURBS, which are equal for the geometry and the numerical analysis (Nguyen et al, 2015).…”

A Comprehensive Review of Isogeometric Topology Optimization: Methods, Applications and Prospects

“…Later, Gao et al [69] constructed an enhanced density distribution function to develop a new ITO method. In the construction of the density distribution function, two steps are involved: (1) Smoothness: the Shepard function is firstly employed to improve the overall smoothness of the densities pre-defined at control points.…”

“…Liu et al [76] also addressed the stressconstrained topology optimization problem of plane stress and bending of thin plates using the ITO framework, where two stability transformation methods are developed to stabilize the optimization using the P-norm function for global stress constraint. Later, Gao et al [77] proposed a NURBS-based Multi-Material Interpolation (N-MMI) model in the ITO method [69] to develop a Multi-material ITO (M-ITO) method. Then Gao et al [78] employed the ITO method to study the design of auxetic metamaterials and the M-ITO method to discuss the optimization of auxetic composites, where a series of novel and interesting material microstructures with the auxetic property can be found.…”

“…[78], which used the SIMPbased ITO method proposed in Ref. [69] and also numerically implemented the energy-based homogenization method to evaluate the effective macroscopic properties using IGA with the imposing of the periodic boundary formulation on the base material unit cell. A reasonable ITO formulation for auxetic metamaterials with the re-entrant and chiral deformation mechanisms is developed, and several optimized design are shown in Figure 10(a).…”

“…Wang et al [85] developed the level setbased ITO method using NURBS, in which NURBS is firstly applied to parametrize the level set function to represent the structural topology and then construct the solution space in IGA to solve the unknown structural responses. Gao et al [69] also employed NURBS to develop an enhanced density distribution function with the sufficient smoothness and continuity to represent the structural topology, and the same NURBS basis functions are also used in IGA. Hou et al [96] used NURBS to construct the MMCs for the representation of the geometries of structural components (a subset of the design domain) with use of explicit design parameters, and the NURBS-based IGA is also applied to solve the structural responses.…”