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Abstract Material interface-related uncertainties induced by inter-diffusion or reactions between two different materials may deteriorate the actual performance of a structural design achieved by topology optimization. Thus a rational methodology is needed to address this issue in the design of hybrid-material engineering products implemented by some novel fabrication techniques such as additive manufacturing. This paper presents a robust shape and topology optimization method accounting for uncertain graded interface properties of multi-material structures. A level set function is used to track the evolving material interfaces during the optimization process, and the material interface uncertainties is modeled by introducing an intermediate zone with graded properties represented by a random field. On the basis of discretizing the input random field by means of the Expansion Optimal Linear Estimation (EOLE) method, the uncertain propagation analysis is implemented with the Polynomial Chaos expansion (PCE) to predict the stochastic response. Then the robust shape and topology optimization problem is stated as a multi-criteria optimization problem, in which the expected value and the standard deviation of the performance function of interest are to be minimized under a given material volume constraint. The shape derivative of the stochastic response is derived in the context of Eulerian description, and then used to advance the evolution of the level set function through the Hamilton-Jacobi equation. In the numerical examples, the proposed robust design method is exemplified by the mean compliance minimization problems. [ABSTRACT FROM AUTHOR]

It is often highly desirable to simultaneously optimize the layout of embedded functional components and the topology of the host structure supporting these components to achieve the best overall performance while still ensuring the structural integrity. We propose a topology optimization framework to account for connecting interface behaviors between the components and the host structure. Here we treat the connecting interfaces with the cohesive zone model to reflect the adhesively bonded interface behaviors. A conforming mesh in conjunction with interface elements is employed to discretize the evolving structure while accounting for the strong discontinuity of displacement field across the material interfaces. To give a clear representation of structural boundaries and the connecting interfaces, we also suggest a multi-material interpolation model in the level set framework, which can conveniently define the connecting interface locations and describe multi-material distribution without redundant phase in the design domain. The objective function is defined as the sum of the strain energy and the work done by the traction on the connecting interface, and the evolution velocities of the level set and the embedded components are treated as design variables. These design variables are updated with the MMA optimizer on the basis of adjoint-variable sensitivity analysis. This optimization formulation allows multiple constraints and mixed design variables ( i.e level set design variables and components’ design variables) to be easily handled in level set based optimization. The design is advanced by the Hamilton–Jacobi equation with the velocity design variables as input. Numerical examples demonstrate the validity and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we propose a new implementation of the level set shape and topology optimization, the velocity field level set method. Therein, the normal velocity field is constructed with specified basis functions and velocity design variables defined on a given set of points that are independent of the finite element mesh. A general mathematical programming algorithm can be employed to find the optimal normal velocities on the basis of the sensitivity analysis. As compared with conventional level set methods, mapping the variational boundary shape optimization problem into a finite‐dimensional design space and the use of a general optimizer makes it more efficient and straightforward to handle multiple constraints and additional design variables. Moreover, the level set function is updated by the Hamilton‐Jacobi equation using the normal velocity field; thus, the inherent merits of the implicit representation is retained. Therefore, this method combines the merits of both the general mathematical programming and conventional level set methods. Integrated topology optimization of structures with embedded components of designable geometries is considered to show the capability of this method to deal with general design variables. Several numerical examples in 2D or 3D design domains illustrate the robustness and efficiency of the method using different basis functions. [ABSTRACT FROM AUTHOR]

Coated structures are commonly used in engineering. The coating material covers the surface of the substrate for protection or to improve certain functionalities. The rising of novel manufacture techniques enables higher design flexibility for such coated structures. This paper presents a level set-based topology optimization method for the design of structures with coating layers. Though a coated structure is composed of two-phase materials, only one level set function is needed in the special case of coating with uniform thickness to describe the distribution of the substrate and the coating layer, thanks to its signed distance property. Without using any intermediate design variables, the proposed method provides a direct interface description between different material phases and geometrical information regarding the coating layer thickness, thus facilitating the sensitivity analysis and numerical implementation. Numerical examples show that the method can be applied to both 2D and 3D problems. [ABSTRACT FROM AUTHOR]

This paper presents a level set-based shape and topology optimization method for conceptual design of cast parts. In order to be successfully manufactured by the casting process, the geometry of cast parts should satisfy certain moldability conditions, which poses additional constraints in the shape and topology optimization of cast parts. Instead of using the originally point-wise constraint statement, we propose a casting constraint in the form of domain integration over a narrowband near the material boundaries. This constraint is expressed in terms of the gradient of the level set function defining the structural shape and topology. Its explicit and analytical form facilitates the sensitivity analysis and numerical implementation. As compared with the standard implementation of the level set method based on the steepest descent algorithm, the proposed method uses velocity field design variables and combines the level set method with the gradient-based mathematical programming algorithm on the basis of the derived sensitivity scheme of the objective function and the constraints. This approach is able to simultaneously account for the casting constraint and the conventional material volume constraint in a convenient way. In this method, the optimization process can be started from an arbitrary initial design, without the need for an initial design satisfying the cast constraint. Numerical examples in both 2D and 3D design domain are given to demonstrate the validity and effectiveness of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

When geometric uncertainties arising from manufacturing errors are comparable with the characteristic length or the product responses are sensitive to such uncertainties, the products of deterministic design cannot perform robustly. This paper presents a new level set-based framework for robust shape and topology optimization against geometric uncertainties. We first propose a stochastic level set perturbation model of uncertain topology/shape to characterize manufacturing errors in conjunction with Karhunen-Loève (K-L) expansion. We then utilize polynomial chaos expansion to implement the stochastic response analysis. In this context, the mathematical formulation of the considered robust shape and topology optimization problem is developed, and the adjoint-variable shape sensitivity scheme is derived. An advantage of this method is that relatively large shape variations and even topological changes can be accounted for with desired accuracy and efficiency. Numerical examples are given to demonstrate the validity of the present formulation and numerical techniques. In particular, this method is justified by the observations in minimum compliance problems, where slender bars vanish when the manufacturing errors become comparable with the characteristic length of the structures. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents a novel topology optimization method for designing structures with multiphase embedded components under minimum distance constraints in the level set framework. By using the level set representation for both the component layout and the host structure topology, the shapes of the components can be easily preserved, and optimal structural topologies with smooth boundary/material interface can be obtained. With the purpose of preventing the components moving too close to each other, a minimum distance constraint based on virtual boundary offset is proposed. Different from existing distance detection methods relying on explicit topology representation, the proposed constraint is imposed as a unified integral form, for which the design sensitivity can be readily obtained. Moreover, this constraint is effective for detecting the distance between any complex-shaped components. Several numerical examples are presented to demonstrate the validity and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a novel topology description model for topology optimization problems with embedded movable holes, combining the ability of the level set model for accurate geometrical description of the prescribed hole shapes, and the high efficiency of the material density-based method. By this means, any arbitrary hole shapes can be represented accurately and smoothly, while the topological changes can be easily handled by the material distribution model. An explicit mathematical expression is defined to obtain the actual structural layout by combing both models. Moreover, an effective model for the non-overlap constraint is proposed in a unified and systematic manner to avoid the overlaps between the holes and between each hole and the design domain boundary. Therein, the non-overlap constraint for all the embedded holes is imposed as a single explicit integral constraint over the design domain, thus avoiding the difficulties in the overlap detection of multiple arbitrary-shaped geometries. Such a non-overlap constraint is accurate and differentiable, facilitating an analytical design sensitivity analysis. Numerical examples are given to demonstrate the effectiveness and efficiency of the present method. [Copyright &y& Elsevier]

This paper presents a level set-based topology optimization method considering the overhang constraint in additive manufacturing (AM) processes. Though the combination of the topology optimization and AM shows a promising potential and high design flexibility, there are still certain limitations. The overhang constraint is one of the major issues that need to be considered in the design stage. It requires the inclination angles of structural downward-facing surfaces to be larger than a given lower bound, so as to prevent the structure from warping or collapsing during the AM process. We propose a new form of overhang constraint in the level set framework, which is expressed as a single domain integral instead of point-wise constraints. This domain integral form facilitates the detection of overhang constraint violation. The shape derivative of the overhang constraint is derived by using the signed distance property of the level set function. The proposed method is capable of dealing with constraints with different minimum overhang angles. Theoretically, it allows the optimization to proceed from an arbitrary structural layout, without the need to satisfy the overhang constraint in the initial design. Several numerical examples are given to show the validity and effectiveness of the proposed method. It is seen in these examples that the overhang constraint is satisfied mainly by adjusting the local shape of structural members violating the overhang constraint during the optimization process. Thus, the overhang angle constrained optimization can generate similar load paths as in conventional optimal designs in most cases, without significantly worsening the structural stiffness. [ABSTRACT FROM AUTHOR]

In most of the existing topology optimization studies of multi-material structures, the interface of different materials was assumed to be perfectly bonded. Optimal design based on the perfect-interface assumption may introduce the risk of failure caused by interface debonding. This paper presents an efficient multi-material topology optimization strategy for seeking the optimal layout of structures considering the cohesive constitutive relationship of the interface. Based on the color level set method to describe the topology and the interface, the interface behavior is simulated by combining the extended finite element method (XFEM) and the cohesive model on fixed mesh. This enables modeling of possible separation of material interfaces, and thus provides a more realistic model of multi-material structures. Furthermore, this interface modeling technique avoids the difficulty of re-meshing when tracking the moving cohesive interface positions during the optimization process. In the topology optimization model, the normal velocities defined on the level set points are considered as design variables. In conjunction with the adjoint-variable sensitivity analysis, these design variables are updated by using the mathematical programming approach and then used to interpolate the boundary velocities. These boundary velocities are extrapolated to the whole domain with the fast marching method and used to advance the structural boundary through the Hamilton–Jacobi equation. This topology optimization technique can handle multiple constraints easily in the framework of level set method and at the same time preserve the signed distance property of the level set functions. Two numerical examples are given to demonstrate the effectiveness of the present method. It is also revealed that the optimal design considering interface behavior may exhibit tension/compression non-symmetric topology, in which material interfaces mainly undergo compression. [ABSTRACT FROM AUTHOR]

This paper proposes a new Multi-Material Level Set (MM-LS) topology description model for topology and shape optimization of structures involving multiple materials. Each phase is represented by a combined formulation of different level set functions. With a total number of M level set functions, this approach provides a representation of M materials and one void phase (totally M + 1 phases). The advantages of the proposed method include: (1) it can guarantee that each point contains exactly one phase, without overlaps between each two phases and redundant regions within the design domain; (2) it possesses an explicit mathematical expression, which greatly facilitates the design sensitivity analysis; and (3) it retains the merits of the level set method, including smooth boundary and distinct interface. A parametric level set method is applied to evolve the topology and shape of multi-material structures, with a high computational efficiency. Several numerical examples are presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

For multi-material structures, ensuring material interface strength is particularly vital for their integrity and durability. In the present study, we incorporate material interfacial stress constraints into topology optimization of bi-material structures. A multi-material level set method, in conjunction with interface-conforming finite element meshes, is employed to describe the distribution of different material phases and to capture the evolution of the material interfaces. The use of interface-conforming meshes enables accurate analysis of both interfacial stresses and their design sensitivities. Noting that the interfacial strength failure is usually characterized by a tension/compression asymmetric mechanism, we adopt an equivalent interfacial stress (which was originally proposed for strength criterion concerning composite delamination) expressed by the interface tensile and tangential stresses in the considered strength criterion. To handle the local nature of such interfacial stress constraints, we propose a global stress measure, which is an approximation of the p-norm of the equivalent interfacial stress field. An adjoint sensitivity analysis scheme is derived by taking into account the interface transmission conditions. To treat multiple constraints (the volume constraints and the interfacial stress constraint) in level set-based topology optimization, the velocity field-level set method is employed. Numerical examples are presented to show effectiveness of the present method. It is also shown that the tension/compression asymmetric interfacial strength criteria may lead to asymmetric designs. • A TO method to constrain interfacial stresses in multi-material structures is proposed. • A globalized tension/compression asymmetric interfacial strength criterion is adopted. • Body-fitted mesh is used to provide accurate interfacial stress results. • Interface transmission conditions are considered in the sensitivity analysis. • The velocity-field level set method is employed to handle multiple TO constraints. [ABSTRACT FROM AUTHOR]