Two-scale Topology Optimization Research Articles

A Two-scale Topology Optimization Method for Functionally Graded Lattice Structures using Three Families of Micro-Structures

A Two-scale Topology Optimization Method for Functionally Graded Lattice Structures using Three Families of Micro-Structures

Two-scale topology optimization method combined with fast non-dominated sorting genetic algorithm II

Two-scale topology optimization method combined with fast non-dominated sorting genetic algorithm II

Stress-constrained concurrent two-scale topology optimization of functionally graded cellular structures using level set-based trimmed quadrilateral meshes

Stress-constrained concurrent two-scale topology optimization of functionally graded cellular structures using level set-based trimmed quadrilateral meshes

Two-scale topology optimization with heterogeneous mesostructures based on a local volume constraint

A new approach to produce optimal porous mesostructures and at the same time optimizing the macro structure subject to a compliance cost functional is presented. It is based on a phase-field formulation of topology optimization and uses a local volume constraint (LVC). The main novelty is that the radius of the LVC may depend both on space and a local stress measure. This allows for creating optimal topologies with heterogeneous mesostructures enforcing any desired spatial grading and accommodating stress concentrations by stress dependent pore size. The resulting optimal control problem is analysed mathematically, numerical results show its versatility in creating optimal macroscopic designs with tailored mesostructures.

Two-Scale Topology Optimization with Isotropic and Orthotropic Microstructures

Advances in additive manufacturing enable the fabrication of complex structures with intricate geometric details, which bring opportunities for high-resolution structure design and transform the potential of functional product development. However, the increasingly delicate designs bring computational challenges for structural optimization paradigms, such as topology optimization (TO), since the design dimensionality increases with the resolutions. Two-scale TO paves an avenue for high-resolution structural design to alleviate this challenge. This paper investigates the efficacy of introducing function-based microstructures into the two-scale TO. Both isotropic and orthotropic microstructure are considered to develop this TO framework. Implicit functions are exploited to model the two classes of cellular materials, including triply periodic minimal surfaces (TPMS) and Fourier series-based functions (FSF). The elasticity tensor of microstructures is computed with numerical homogenization. Then, a two-scale TO paradigm is formulated, and a gradient-based algorithm is proposed to simultaneously optimize the micro-scale structures and macro-scale material properties. Several engineering benchmark cases are tested with the proposed method, and experimental results reveal that using proposed microstructures leads to, at most, a 36% decrease in the compliance of optimal structures. The proposed framework provides achievable directionality and broader design flexibility for high-resolution product development.

Two-scale topology optimization for transient heat analysis in porous material considering the size effect of microstructure

Two-scale topology optimization for transient heat analysis in porous material considering the size effect of microstructure

Combining Macro- and Mesoscale Optimization: A Case Study of the General Electric Jet Engine Bracket

Topology optimization (TO) is a mathematical method that optimizes the material layout in a pre-defined design domain. Its theoretical background is widely known for macro-, meso-, and microscale levels of a structure. The macroscale TO is now available in the majority of commercial TO software, while only a few software packages offer a mesoscale TO with the design and optimization of lattice structures. However, they still lack a practical simultaneous macro–mesoscale TO. It is not clear to the designers how they can combine and apply TO at different levels. In this paper, a two-scale TO is conducted using the homogenization theory at both the macro- and mesoscale structural levels. In this way, the benefits of the existence and optimization of mesoscale structures were researched. For this reason, as a case study, a commercial example of the known jet engine bracket from General Electric (GE bracket) was used. Different optimization workflows were implemented in order to develop alternative design concepts of the same mass. The design concepts were compared with respect to their weight, strength, and simulation time for the given load cases. In addition, the lightest design concept among them was identified.

Data-driven topology optimization of spinodoid metamaterials with seamlessly tunable anisotropy

We present a two-scale topology optimization framework for the design of macroscopic bodies with an optimized elastic response, which is achieved by means of a spatially-variant cellular architecture on the microscale. The chosen spinodoid topology for the cellular network on the microscale (which is inspired by natural microstructures forming during spinodal decomposition) admits a seamless spatial grading as well as tunable elastic anisotropy, and it is parametrized by a small set of design parameters associated with the underlying Gaussian random field. The macroscale boundary value problem is discretized by finite elements, which in addition to the displacement field continuously interpolate the microscale design parameters. By assuming a separation of scales, the local constitutive behavior on the macroscale is identified as the homogenized elastic response of the microstructure based on the local design parameters. As a departure from classical FE2-type approaches, we replace the costly microscale homogenization by a data-driven surrogate model, using deep neural networks, which accurately and efficiently maps design parameters onto the effective elasticity tensor. The model is trained on homogenized stiffness data obtained from numerical homogenization by finite elements. As an added benefit, the machine learning setup admits automatic differentiation, so that sensitivities (required for the optimization problem) can be computed exactly and without the need for numerical derivatives – a strategy that holds promise far beyond the elastic stiffness. Therefore, this framework presents a new opportunity for multiscale topology optimization based on data-driven surrogate models.

Topology optimization of phononic-like structures using experimental material interpolation model for additive manufactured lattice infills

Phononic crystals (PnCs) have seen increasing popularity due to band gap property for sound wave propagation. As a natural bridge, topology optimization has been applied to the design of PnCs. However, thus far most of the existent works on topological design of PnCs have been focused on single micro-scale topology optimization of a periodical unit cell. Moreover, practical manufacturing of those designed structures has been rarely involved. This paper presents a quasi two-scale topology optimization framework suitable for additive manufacturing (AM) implementation to design 2D phononic-like structures with respect to sound transmission coefficient (STC). A designate topology is employed and subjected to sizing optimization in the micro-scale design. The thin-walled square lattice structures made of single metal material are selected as the infills for the design domain to guarantee material connectivity in the optimized design in order to facilitate fabrication by AM. The practical effective mechanical property of the lattice structures with different volume densities obtained by experimental measurement is employed in the topology optimization. The proposed framework is applied to the design of 2D phononic-like structures with different macroscopic shapes for the desired band gap feature. Numerical examples show the desired band gap containing a prescribed excitation frequency can be realized through the proposed quasi two-scale topology optimization method. Moreover, the optimized designs are reconstructed into CAD files with the thin-walled lattice infills. The reconstruction makes fabrication of the optimized designs feasible by practical AM process.

A Two-Scale Multi-Resolution Topologically Optimized Multi-Material Design of 3D Printed Craniofacial Bone Implants.

Bone replacement implants for craniofacial reconstruction require to provide an adequate structural foundation to withstand the physiological loading. With recent advances in 3D printing technology in place of bone grafts using autologous tissues, patient-specific additively manufactured implants are being established as suitable alternates. Since the stress distribution of these structures is complicated, efficient design techniques, such as topology optimization, can deliver optimized designs with enhanced functionality. In this work, a two-scale topology optimization approach is proposed that provides multi-material designs for both macrostructures and microstructures. In the first stage, a multi-resolution topology optimization approach is used to produce multi-material designs with maximum stiffness. Then, a microstructure with a desired property supplants the solid domain. This is beneficial for bone implant design since, in addition to imparting the desired functional property to the design, it also introduces porosity. To show the efficacy of the technique, four different large craniofacial defects due to maxillectomy are considered, and their respective implant designs with multi-materials are shown. These designs show good potential in developing patient-specific optimized designs suitable for additive manufacturing.

Study the Effect of Microstructures with Anisotropic Mechanical Properties in Two-Scale Topology Optimization

Study the Effect of Microstructures with Anisotropic Mechanical Properties in Two-Scale Topology Optimization

Universal machine learning for topology optimization

We put forward a general machine learning-based topology optimization framework, which greatly accelerates the design process of large-scale problems, without sacrifice in accuracy. The proposed framework has three distinguishing features. First, a novel online training concept is established using data from earlier iterations of the topology optimization process. Thus, the training is done during, rather than before, the topology optimization. Second, a tailored two-scale topology optimization formulation is adopted, which introduces a localized online training strategy. This training strategy can improve both the scalability and accuracy of the proposed framework. Third, an online updating scheme is synergistically incorporated, which continuously improves the prediction accuracy of the machine learning models by providing new data generated from actual physical simulations. Through numerical investigations and design examples, we demonstrate that the aforementioned framework is highly scalable and can efficiently handle design problems with a wide range of discretization levels, different load and boundary conditions, and various design considerations (e.g., the presence of non-designable regions).

Two-scale design of porosity-like materials using adaptive geometric components

This paper is an extension of our recent work that presents a two-scale design method of porosity-like materials using adaptive geometric components. The adaptive geometric components consist of two classes of geometric components: one describes the overall structure at the macrostructure and the other describes the structure of the material at the microstructures. A smooth Heaviside-like elemental-density function is obtained by projecting these two classes on a finite element mesh, namely fixed to reduce meshing computation. The method allows simultaneous optimization of both the overall shape of the macrostructure and the material structure at the micro-level without additional techniques (i.e., material homogenization), connection constraints, and local volume constraints, as often seen in most existing methods. Some benchmark structural design problems are investigated and a selected design is post-processed for 3D printing to validate the effectiveness of the proposed method.
 Keywords:
 topology optimization; concurrent optimization; porosity structures; two-scale topology optimization; adaptive geometric components.

Two-Scale Topology Optimization of the 3D Plant-Inspired Adaptive Cellular Structures for Morphing Applications

AbstractA novel two-scale topology optimization method is developed in this work to optimize three-dimensional (3D) plant-inspired fluidic adaptive cellular structures for morphing applications. In...

Two-scale buckling topology optimization for grid-stiffened cylindrical shells

Abstract Stiffened shells are widely used in aerospace structures such as launch vehicles and aircraft wings. This paper presents a novel two-scale topology optimization method to design the innovative grid-stiffened pattern for maximizing the critical buckling load of thin-walled cylindrical shells. On the micro-scale, the asymptotic homogenization method is employed to calculate the general stiffness coefficients of the cell. On the macro-scale, the maximum critical buckling load is set as the objective to drive the topology optimization on the micro-scale. Besides, the repeated eigenvalues are considered both in the sensitivity analysis of buckling loads and the optimization solver with a sub-problem based on the Method of Moving Asymptotes (MMA). Through the optimization, we can obtain the optimal configuration of the grid-stiffened cell. In this paper, an illustrative example of the grid-stiffened cylindrical shell for maximizing the critical buckling load is carried out to validate the effectiveness of the proposed optimization method. In comparison to the optimal orthogrid shell, the critical buckling load with the optimal pattern obtained by the proposed optimization method have a dramatically increase of 21.7%. It can be concluded that the proposed method has huge potential to design the configuration of the grid-stiffened cell of cylindrical shells.

Design of phononic-like structures and band gap tuning by concurrent two-scale topology optimization

Phononic crystals have been paid plenty of attention due to the particular characteristics of band gap for elastic wave propagation. Many works have been focused on the design of the phononic crystals materials/structures through different methods including experimental and numerical approaches such as topology optimization. However, most of the works on topological design of the phononic materials/structures are on micro-scale topology optimization of the crystal unit cell based on the assumption of infinite periodicity. Finite design domain and corresponding boundary condition are seldom considered directly in the single micro-scale topology optimization of the crystal unit cell. This paper presents a concurrent two-scale topology optimization framework to design phononic-like structures with respect to the vibro-acoustic criterion, and the finite dimension and the boundary condition of the macro-scale design domain can be fully taken into consideration simultaneously. Accuracy of the proposed model and method to compute the wave band gap property of two-dimensional phononic structures is validated. Then the concurrent two-scale topology optimization approach is employed to design the phononic-like structures and tune the wave band gap property. Numerical examples show the advantage of the concurrent two-scale topology optimization over the single micro-scale design of the crystal unit cell. Many interesting features of the proposed approach are also revealed and discussed. The presented work shows that the concurrent two-scale topology optimization approach is promising to be a powerful tool in the design of vibro-acoustic phononic-like structures for achieving the desired band gap property.

Two-scale topology optimization using homogenization theory

Homogenization theory forms the basis for solving the topology optimization problem (TOP) of composite structures. The simplest repeating unit of the microstructure, that if isolated represents exactly the macroscopic behavior of the structure, is called the unit cell. Scope of homogenization is to determine the macroscopic properties of the non-homogeneous unit cell. In this study, homogenization is implemented on a 3D lattice unit cell, with the radius of the unit cell being the varying parameter of the homogenization procedure. Different values of the radius result to different configurations, hence, to different equivalent properties of the unit cell. Therefore, a fitting process takes place in order to accurately model the variations of the obtained effective properties with respect to the design variable. The corresponding, homogenization-based TOP is posed and the resulting geometries for several case studies are presented.

Gradient-Based Two-Scale Topology Optimization With B-Splines on Sparse Grids

Structural optimization searches for the optimal shape and topology of components such that specific physical quantities are optimized, for instance, the stability of the resulting structure. Probl...

Multiscale design for additive manufactured structures with solid coating and periodic infill pattern

This paper considers multiscale topology optimization of an infill structure. The structure is composed of an optimized layout configuration at the macroscale with uniform optimized microstructures as infill lattices coated by a thin skin. The design optimization of the infill lattice is performed simultaneously with the topology optimization of macroscale structure, which also includes the coating. The classic Solid Isotropic Material with Penalization (SIMP) method is used to design the topology of the microstructure. The Novel Implementation of Asymptotic Homogenization (NIAH) evaluates the effective properties of the microstructure; the macroscale structural analysis uses these homogenized properties. The Porous Anisotropic Material with Penalization (PAMP) method realizes the two-scale topology optimization of the macrostructure and microstructure. Harmonic-mean based non-linear filters identify the coating layer and realize length scale control on the topologies of the macrostructure as well as the microstructure. These morphology mimicking non-linear filters also distinguish the skin and the infill lattice parts of the design at the macroscale. Several numerical examples demonstrate the effectiveness of the proposed optimization method.

Two-scale topology optimization of macrostructure and porous microstructure composed of multiphase materials with distinct Poisson’s ratios

Abstract Negative Poisson’s ratio (NPR) material attracts a lot of attentions for its unique mechanical properties. However, achieving NPR is at the expense of reducing Young’s modulus. It has been observed that the composite stiffness can be enhanced when blending positive Poisson’s ratio (PPR) material into NPR material. Based on the respective interpolation of Young’s modulus and Poisson’s ratio, two concurrent topology optimization problems with different types of constraints, called Problem A and B, are respectively discussed to explore the Poisson’s ratio effect in porous microstructure. In Problem A, the volume constraints are respectively imposed on macro and micro structures; in Problem B, besides setting an upper bound on the total available base materials, the micro thermal insulation capability is considered as well. Besides considering the influence of micro thermal insulation capability on the optimized results in Problem B, the similar and dissimilar influences of Poisson’s ratios, volume fractions in Problem A and B are also investigated through several 2D and 3D numerical examples. It is observed that the concurrent structural stiffness resulting from the mixture of PPR and NPR base materials can exceed the concurrent structural stiffness composed of any individual base material.