Topological optimization plays a guiding role in the conceptual design process. This paper conducts research on structural topology optimization algorithm within the framework of isogeometric analysis. For multi-component structures, the Nitsche’s method is used to glue different meshes to perform isogeometric multi-patch analysis. The discrete variable topology optimization algorithm based on integer programming is adopted in order to obtain clear boundaries for topology optimization. The sensitivity filtering method based on the Helmholtz equation is employed for averaging of curved elements' sensitivities. In addition, a simple averaging method along coupling interfaces is proposed in order to ensure the material distribution across coupling areas is reasonably smooth. Finally, the performance of the algorithm is demonstrated by numerical examples, and the effectiveness of the algorithm is verified by comparing it with the results obtained by single-patch and ABAQUS cases.

Topology optimization of structures aims to find reasonable material distribution under certain objectives. Combined with Finite Element Method (FEM), by utilizing powerful computational resources, nowadays topology optimization is widely applied for engineering problems. As a new generation of FEM, Isogeometric Analysis (IGA) [

In 2008, Wall et al. developed a shape optimization strategy by moving control points [

Since in engineer practice a structure model is usually made upon NURBS patches of parts, in this research we mainly study topology optimization of multi-patch IGA, especially in 2D. In FEM framework, Jeong et al. used a condensed mortar method to glue dissimilar interfaces for topology optimization and achieved good performance [

In this paper, we adopt the Nitsche’s method for NURBS patch coupling, by adding external stiffness by multiplying the averaging force and displacement gap along coupling interfaces. The Nitsche’s method introduces no additional DoFs, thus no special treatment is needed for the system, which makes the method suitable for contact problems [

The structure of this paper is organized as follows. In

FEM analysis needs two sets of basis functions, one is used for geometry model building, the other one is used for displacement field construction. The main idea of IGA consists in using NURBS as basis functions for both geometry models and the displacement field. The present research mainly focuses on 2D structures, thus only 2D surfaces are introduced, however 3D volumes can be naturally constructed by adding one more dimension.

A 2D surface, denoted by

Following the IGA concept, we approximate the displacement field by

As illustrated in

Consider an interface problem, in which the domain

The Nitsche parameter

for

for

for

In this research, we set

From the matrix point of view, imagine we have the original discretized IGA formulation in matrix form, as in

As illustrated in

In cases of 2D linear elasticity, we have

Consider a minimization problem of structural compliance

In the algorithm proposed by Liang et al. [

It should be noted that the density design variable

After filtering the sensitivities of the design variables from

For

Therefore the original design variable

In order to ensure the above dual problem has critical dual variables that are strictly larger than zero, and thereby to ensure the positive definiteness of the above optimization problem, the objective function of

Apparently we have

The presented algorithm is composed by

Due to the integer programming sub-problems are constructed by sensitivity information, the accuracy of these sub-problems is only valid within a local range of current design variables. In order to ensure the approximation accuracy of the integer programming sub-problems, and to make the solutions of the sub-problems converge slowly to the solutions of the original problem, a moving limit strategy should be adopted. Specifically, we gradually reduce the material consumption constraint by a volume fraction reduction factor

For the present topology optimization technique by the sequential integer programming method, by setting the element density as 1 or 0 in the parameter space, we could achieve a black-white topology design accordingly in the physical space with clear and curved boundaries, as shown in

In IGA, although the meshes in the parameter space are structured, however the meshes in the physical space are usually curved after mapping. The classical sensitivity filtering technique needs to search adjacent elements and then perform averaging methods, this is not suitable for complex curved meshes in IGA. In this study, the implicit filtering method based on Helmholtz equation [

The transformation matrix is given as

Finally, the filtering formula is obtained as

Note that

The flow chart of the discrete variable topology optimization algorithm is drawn in

However, for multi-patch sensitivities, since the mesh resolutions of different patches are usually different, the magnitudes of the sensitivity values correspond to meshes of different densities are also different. If these sensitivities are directly substituted into the topology optimization algorithm, it could lead to inappropriate optimization results, especially along the coupling interfaces. In this paper, a simple sensitivity filtering method for the coupling interface is proposed. Suppose that the numbers of elements along both sides of the coupling interface is

Correspondingly, the sensitivities of both sides is scaled as

Since the meshes on both sides are in the same level at this stage, sensitivity filtering can be performed by using a simple averaging method, and the filtered sensitivities

Finally, the sensitivity is restored to the original meshes by adding all the values within each element

Note that although the relationship between the sensitivities and the mesh resolutions is not linear, we still scale the sensitivities linearly according to mesh refinement by

In the following examples, the material of Young’s modulus

In order to show the performance of the method in terms of coupling and optimization, and to compare with the existing result of the Mortar method [

Compute this example using IGA with the Nitsche’s method to glue the overlapped interface, we plot the displacement distribution in

Moreover, the curve about the compliance in

As the second example, consider the cantilever beam problem as shown in

The contour plot of displacement distribution by multi-patch IGA is shown in

In order to illustrate the effectiveness of the proposed method, the result by our method is compared with the results by single patch (

The mathematical nature of the topology optimization problem with discrete variables consists in nonlinear integer programming with partial differential equation constraints, and different discretizations will lead to different local optimal solutions. The method proposed in this paper provides a feasible method for mechanical analysis and topology optimization under different meshes. In practical engineering, structures are usually made upon parts assembling, the advantage of using multi-patches of NURBS models is that it allows engineers to use independent meshes of different parts for analysis and design, which could greatly reduce the mesh requirements.

Finally consider the one-quarter annulus problem as shown in

The displacement distribution of the initial configuration is drawn in

The contour plot of the Mises stress is shown in

For multi-patch models in engineer practices, it is necessary to analyze and optimize from the perspective of the entire structure. In this paper, the Nitsche’s method is used to perform the isogeometric multi-patch analysis, which allows the patches of independent meshes to be glued with each other, thus greatly reducing the quality requirements for mesh generation. Furthermore, the discrete topology optimization method based on integer programming is introduced to obtain black-white boundaries of the conceptual design. Taking advantages of curved-edge elements in IGA, topology optimization in the framework of IGA can obtain locally smoother material boundaries than traditional FEM under the same level of mesh resolution. In addition, a sensitivity filtering method based on the Helmholtz equation is used for complex curved-edge elements, which not only ensures the filtering effect, but also saves the computational resources for searching of adjacent elements. In addition, for the coupling interfaces, an averaging method of interface sensitivities is proposed to ensure the transition smoothness of materials. Finally the performance of the proposed method is tested and verified by numerical examples.

It should be noted that although the topology optimization results in this research is totally black and white, the resulting design is still unavoidably jagged due to the nature of the IGA meshes and the nature of the adopted pixel-based element-wise optimization method. However, we believe that the research results have good potential to be adopted as initial configurations for some optimization methods, for instance to provide initial configuration guesses for MMC [

The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.