Continuum Topology Optimization Research Articles

Efficient Acoustic Topology Optimization Using Vibro-Acoustic Coupled Craig–Bampton Mode Synthesis

Continuum topology optimization is an effective way to reduce vibration and noise of vibro-acoustic coupling system. However, the high computation time required for calculating the vibro-acoustic responses during topology optimization is a major obstacle for practical applications. In this paper, a novel symmetric method of vibro-acoustic system matrix is proposed, and a new model reduction method is developed based on Craig–Bampton mode synthesis method to compute dynamic responses with adequate efficiency and accuracy for topology optimization. The comparison results show that the proposed method substantially reduces the degrees of freedom (DOFs) and calculation time. The response values and eigenfrequencies calculated by the model reduction method are exactly the same as those of the full model. Furthermore, the bidirectional evolutionary structural optimization (BESO) algorithm is applied to solve the problem of minimizing the response of a single frequency and a certain range of frequency excitation at a specified target point. The results show that the optimization algorithm converges fast, the iterative process is robust and the response values of different coupling systems can be reduced to a higher extend, which indicates the applicability of the optimization algorithm.

Discrete topology optimization in augmented space: integrated element removal for minimum size and mesh sensitivity control

A computational strategy is proposed to circumvent some of the major issues that arise in the classical threshold-based approach to discrete topology optimization. These include the lack of an integrated element removal strategy to prevent the emergence of hair-like elements, the inability to effectively enforce a minimum member size of arbitrary magnitude, and high sensitivity of the final solution to the choice of ground structure. The proposed strategy draws upon the ideas used to arrive at mesh-independent solutions in continuum topology optimization and enables efficient imposition of a minimum size constraint onto the set of non vanishing elements. This is achieved via augmenting the design variables by a set of auxiliary variables, called existence variables, that not only prove very effective in addressing the aforementioned issues but also bring in a set of added benefits such as better convergence and complexity control. 2D and 3D examples from truss-like structures are presented to demonstrate the superiority of the proposed approach over the classical approach to discrete topology optimization.

On the Six Node Hexagon Elements for Continuum Topology Optimization of Plates Carrying in Plane Loading and Shell Structures Carrying out of Plane Loading

The need of polygonal elements to represent the domain is gaining interest among structural engineers. The objective is to perform static analysis and topology optimization of a given continuum domain using the rational fraction type shape functions of six node hexagonal elements. In this paper, the main focus is to perform the topology optimization of two-dimensional plate structures using Evolutionary Swarm Intelligence Firefly Algorithms (ESIFA) and three-dimensional shell structures using optimality criteria. The optimization of plates carrying in plane loading is performed with minimum weight as objective. Two different types of shell structures are optimized using maximum strain energy as criteria. The optimal distribution of the material in the design domain obtained using six node hexagon elements is compared with the optimal distribution of material obtained using quadrilateral elements. A few problems from the literature have been solved and this study has proved that hexagon element gives better results over traditional quadrilateral elements.

Automated optimization-based generation and quantitative evaluation of Strut-and-Tie models

Strut-and-Tie modelling (STM) is a well-known approach to design D-regions in reinforced concrete structures. Because the STM method is based on lower-bound analysis, finding a suitable truss-analogy model is the most important aspect to guarantee good structural and economic performance of a resulting design. Continuum topology optimization (TO) methods have been studied for two decades to solve this problem. However, while these studies provide inspiration to designers, they lack the capability to automatically generate valid truss-analogy models as needed in the STM method. In order to prevent manual interpretation and automatically generate suitable STM models for various D-regions, a method is proposed for the generation of optimization-based STM (OPT-STM) models. The proposed method includes three phases: the TO phase, the topology extraction phase and the shape optimization phase. Next, in order to evaluate the effectiveness of the generated OPT-STM models, an evaluation using Nonlinear Finite Element Analysis (NLFEA) is performed to analyze the performance of STM models. For two D-region problems, two OPT-STM models and 11 manually constructed STM models from literature are evaluated, and their performance is compared and discussed, demonstrating the validity and effectiveness of the proposed automated generation method.

A novel stress influence function (SIF) methodology for stress-constrained continuum topology optimization

This study presents a new stress influence function (SIF) methodology for continuum topology optimization under consideration of local strength failure. Firstly, the qp-relaxation criterion is involved to circumvent the stress singularity. To deal with the large-scale stress constraints in topology optimization, the local stress constraint is reflected in the objective along with the material volume by multiplication, and the weight of stress is characterized by stress influence function. Meanwhile, three types of stress influence functions are proposed for comparison. By means of the study on the characteristic of high-stress elements, the rationality of the SIF methodology is illustrated, in which the proposed method may achieve the full-stress state of high-stress element. Numerical examples are given to demonstrate the applicability and validity of the proposed methodology. It is shown that the proposed methodology can obtain reasonable results. Consequently, the proposed SIF methodology provides a novel strategy with high computational efficiency for topology optimization considering local strength failure.

Topology optimization of continuum supporting structures for microwave antenna applications

In microwave antenna applications, continuum structures usually support attached functional surfaces to realize some specific electromagnetic performance. Topology optimization of continuum supporting structures with functional surfaces is a challenge for microwave antenna applications. By introducing the concept of aperture field distribution into the design domain, a weighting approach for the topology optimization of continuum supporting structures with functional surfaces is presented based on the SIMP model. With the weighting aperture field distribution, the objective function of compliance in the previous SIMP method is changed to a weighted compliance. By selecting an optimized control factor, a different truss topology structure with several components from the previous method is clearly obtained. The effectiveness of the proposed method is validated through three typical applications: array antennas, reflector antennas, and conformal antennas with planar and curved functional surfaces.

Topology optimization with nozzle size restrictions for material extrusion-type additive manufacturing

Topology optimization that is tailored to additive manufacturing constraints and possibilities is an important area of research with direct implications on solution manufacturability. This paper focuses on implementing the nozzle size constraint that is associated with most material extrusion-type additive manufacturing processes, such as fused filament fabrication and concrete 3D printing. The constraint is especially important for manufacturability in situations where the size of used nozzle is large in comparison with the size of the design domain. This paper suggests a new projection-based algorithm that embeds material extrusion-type primitives into the projection methodology used for material distribution approaches to topology optimization. Projection-based algorithms for continuum topology optimization have received considerable attention in recent years due to their ability to improve manufacturability in a flexible and computationally efficient manner. A formulation for single-directional primitives is presented and extended to allow the use of two-directional primitives that can simulate a more realistic nozzle movement. The proposed algorithms are demonstrated on 2D benchmark problems and are shown to satisfy the imposed nozzle size restrictions.

H-adaptive topology optimization considering variations of material properties and energy error density recovery

PurposeThe purpose of this paper is to propose a new scheme for obtaining acceptable solutions for problems of continuum topology optimization of structures, regarding the distribution and limitation of discretization errors by considering h-adaptivity.Design/methodology/approachThe new scheme encompasses, simultaneously, the solution of the optimization problem considering a solid isotropic microstructure with penalization (SIMP) and the application of the h-adaptive finite element method. An analysis of discretization errors is carried out using an a posteriori error estimator based on both the recovery and the abrupt variation of material properties. The estimate of new element sizes is computed by a new h-adaptive technique named “Isotropic Error Density Recovery”, which is based on the construction of the strain energy error density function together with the analytical solution of an optimization problem at the element level.FindingsTwo-dimensional numerical examples, regarding minimization of the structure compliance and constraint over the material volume, demonstrate the capacity of the methodology in controlling and equidistributing discretization errors, as well as obtaining a great definition of the void–material interface, thanks to the h-adaptivity, when compared with results obtained by other methods based on microstructure.Originality/valueThis paper presents a new technique to design a mesh made with isotropic triangular finite elements. Furthermore, this technique is applied to continuum topology optimization problems using a new iterative scheme to obtain solutions with controlled discretization errors, measured in terms of the energy norm, and a great resolution of the material boundary. Regarding the computational cost in terms of degrees of freedom, the present scheme provides approximations with considerable less error if compared to the optimization process on fixed meshes.

Design and analysis of porous flexure hinge based on dual-objective topology optimization of three-dimensional continuum

This paper properly presents a dual-objective topology optimization mathematical model for compliance and precision performance of flexure hinges based on the three-dimensional continuum topology optimization. The OptiStruct module is applied to solving the problem. And through the post-processing design and parameterization of the results, a single-axis quasi-leaf porous flexure hinge (QLPFH) is generated. The dimensionless empirical equations of the stiffness and rotational center of the hinge are derived by finite element analysis, and the performance of the flexure hinge under different dimensionless parameters is analyzed. Comparing the QLPFH and the leaf flexure hinge (LFH) with the same external space dimensions, the results show that the translational stiffness of the LFH in the desired working direction and the rotational stiffness are much larger than those of the QLPFH. In addition, in the case of the stiffness of the QLPFH is greatly reduced, its precision performance has also been improved to a certain extent. The feasibility of dual-objective design method based on three-dimensional continuum topology optimization is further confirmed.

Improved genetic algorithm with two-level multipoint approximation for complex frame structural optimization

In this paper, an improved structural topology and sizing optimization method is developed for the fast and efficient engineering design of complex frame structures where beam elements are mainly used in the structures. Discrete and continuous variables are included that the elimination or existence of beam elements are treated as discrete variables (0,1), and the continuous sizing variables of beam cross sections are considered to be continuous variables. To solve the mixed variable problem, the paper introduces a two-level multipoint approximation strategy (TMA). The first-level approximate problem is established by using the branched multipoint approximate function, which includes both two types of variables. Genetic algorithm (GA) is used to determine the absence or presence of beam members. The second-level approximate problem that only involving retained continuous size variables is made on this basis, which uses Taylor expansion and dual methods to solve the inner layer continuous optimization problem. Meanwhile, a strategy of adding a new complementary design point is adopted to expend the search scopes and improve the precision. Temporal deletion techniques are used to temporarily remove redundant constraints and local vibration modes processing techniques are used for continuum topology optimization under frequency constraints. Several representative examples are investigated to validate the effectiveness of the improved method.

Efficient multi-material continuum topology optimization considering hyperelasticity: Achieving local feature control through regional constraints

We introduce a general and efficient multi-material topology optimization framework considering hyperelasticity with many local constraints, which enables flexible control of the design’s local features. The proposed framework can effectively distribute multiple candidate materials described by distinct constitutive models according to their respective nonlinear behaviors, and efficiently handle a flexible setting of volume constraints, either global or local. To ensure computational efficiency of the proposed framework, we present a virtual element-based formulation in conjunction with a tailored adaptive refinement and coarsening scheme for multi-material problems, and we adopt the ZPR scheme to update the design variables associated with each constraint in parallel. Three design examples are presented, demonstrating the efficiency and effectiveness of the proposed framework in distributing multiple candidate materials with distinct nonlinear elastic behaviors and handling both global and many (e.g., 1024) local constraints. We envision that the proposed framework enables unique computational capabilities for designing next-generation composite metamaterials and structures with nonlinear behaviors and multi-functionalities.

The robust fail-safe topological designs based on the von Mises stress

For large-scale equipment, e.g. aerospace and architecture industry, it is valuable to guarantee that one structure could survive partial damages. Due to the location of the damage is unknown in prior, results in a high number of failure scenarios to be calculated when considering fail-safe requirement in topology optimization. In this article, we propose an efficient continuum topology optimization method on the basis of the design philosophy of robust fail-safe structure. A number of patches with predefined shapes are used to simulate the material failure. The material properties of damaged models are interpolated by von Mises stress to construct the well-posed optimization model. The damaged compliance for the worst failure case is set as the optimization objective, and the KS function is adopted to approximate the non-differentiable max-operator. A computationally efficient sensitivity formulation is derived via the adjoint method. To suppress the highly nonlinear stress behavior and the phenomenon of optimization oscillation, an extended variable update scheme within the framework of Optimality Criteria (OC) method is developed. Representative benchmarks show that the presented strategy has the effectiveness at yielding the fail-safe structure by using acceptable computational cost.

A novel approach of reliability-based topology optimization for continuum structures under interval uncertainties

Purpose The purpose of this paper is to propose a novel non-probabilistic reliability-based topology optimization (NRBTO) method for continuum structural design under interval uncertainties of load and material parameters based on the technology of 3D printing or additive manufacturing. Design/methodology/approach First, the uncertainty quantification analysis is accomplished by interval Taylor extension to determine boundary rules of concerned displacement responses. Based on the interval interference theory, a novel reliability index, named as the optimization feature distance, is then introduced to construct non-probabilistic reliability constraints. To circumvent convergence difficulties in solving large-scale variable optimization problems, the gradient-based method of moving asymptotes is also used, in which the sensitivity expressions of the present reliability measurements with respect to design variables are deduced by combination of the adjoint vector scheme and interval mathematics. Findings The main findings of this paper should lie in that new non-probabilistic reliability index, i.e. the optimization feature distance which is defined and further incorporated in continuum topology optimization issues. Besides, a novel concurrent design strategy under consideration of macro-micro integration is presented by using the developed RBTO methodology. Originality/value Uncertainty propagation analysis based on the interval Taylor extension method is conducted. Novel reliability index of the optimization feature distance is defined. Expressions of the adjoint vectors between interval bounds of displacement responses and the relative density are deduced. New NRBTO method subjected to continuum structures is developed and further solved by MMA algorithms.

Toward holistic tension- or compression-biased structural designs using topology optimization

The efficiency of a structural design relies considerably on the knowledge and experience of its design engineer. It is critical to choose the proper structural system, cross-sections, and materials as these elements significantly influence the design’s performance. Establishing computational methods that support the engineer in the design process and facilitate more objective solutions therefore represents an important step in creating proficient structural designs.In this paper, a novel approach is introduced to generate topology optimization results with an intended bias toward tension or compression for practical application in structural engineering. The results are deduced from the example of continuum elements and extended to cover truss topology optimization – assuming a linear-elastic, isotropic material model – in addition to combined truss–continuum topology optimization with bilinear, orthotropic material behavior. The effects of creating such solutions are analyzed at a simple theoretical problem. For the purposes of practically designing reinforced concrete structures, the preferable scope of application of each of the three methods is evaluated. The benefit of creating biased solutions is demonstrated by assessing their structural performance, robustness, and overall suitability on both global (system) and local (cross-section) scales and by applying them to common problems in structural engineering.

A mathematical programming method for the topology optimization of a truss-like continuum

A mathematical programming method for the topology optimization of a truss-like continuum

Topology optimization of continua considering mass and inertia characteristics

The center of mass and the inertia tensor are important parameters characterizing the kinetic properties of many structural systems, e.g., the rotor, spacecraft, and aircraft. In these occasions, designs not consistent with the requirements on the center of mass and inertia tensor cannot be accepted. In this paper, mathematical models are established to carry out the topology optimization of continua considering the mass and inertia characteristics. Two numerical examples, one for minimizing the static compliance considering locations of the center of mass and the other for maximizing the fundamental eigenfrequency considering both locations of the center of mass and values of the inertia products, are presented to demonstrate that the requirements on mass and inertia characteristics could have a profound influence on the final optimized layouts in structural topology optimization problems. On the other hand, the mass and inertia characteristics can also be tuned by topology optimization methods. In the second numerical problem, the smooth minimum operator is used to approximate the fundamental eigenfrequency with the first 4 eigenfrequencies so that the possible non-differentiability of the repeated eigenfrequencies can be avoided. Due to its extreme importance in engineering practice, the work of this paper can be useful to carry out designs in accordance with the requirements on the mass and inertia characteristics under the framework of traditional density-based structural topology optimization methods.

Design of buckling constrained multiphase material structures using continuum topology optimization

This study contributes to a possibility of evaluating composite structures configuration such as steel and concrete using buckling and volume constraints based on multi-material topology optimization. A Jacobi active-phase algorithm is used to generate multiphase topology optimization. It provides a rational solution appropriated to the topology optimizer, Method of Moving Asymptotes due to the conflict in updating the design variables. A modified material interpolation scheme solving spurious buckling modes problem which occurs in the multi-material topology optimization process is given and discussed. An investigation of buckling constraint parameter is described. It allows a single-objective minimum compliance topology optimization to obtain two objectives of maximizing both structure stiffness and first buckling load factor. The optimal changing topologies of single material structure and multi-material structure corresponding to different buckling constraints are presented. Numerical examples of compression-only structures and compression-tension structures considering structural instability are performed using both single material and multiple materials to verify the efficiency and superiority of the present method.

Design and analysis of Multicavity Flexure Hinge (MCFH) based on three-dimensional continuum topology optimization

This paper proposes a novel single-axis Multicavity Flexure Hinge (MCFH) consisting of the symmetric cycloidal notches and the multiple cavities. This flexure hinge is firstly designed based on the three-dimensional continuum topology optimization (3-DCTO). And then post-processing is used to obtain the final configuration of the hinge. In order to study the effect of the stiffness performance on different geometric parameters of the hinge, the dimensionless stiffness empirical equations are derived based on finite element analysis (FEA). Subsequently, the new flexure hinge is compared with various types of hinges on the performance. The results show that the presented flexure hinge has greater compliance in the working direction and the results also confirm the 3-DCTO method can provide a new idea for the design of flexure hinges in the future.

An adaptive mesh‐adjustment strategy for continuum topology optimization to achieve manufacturable structural layout

SummaryThis paper presents an adaptive mesh adjustment algorithm for continuum topology optimization method to describe the structural boundary using nonuniform isoparametric element. A criterion on the basis of the node movement is proposed; herein, the densities and coordinates of the nodes are defined to instruct the deformation of finite elements in subsequent optimization iterations. With such a scheme, the topology optimization can start from a regular mesh discretization then gradually yields an optimal design with clear structural boundaries. The element in the transition along the boundary is refined; on the contrary, the pure solid or void element is coarsen. The contribution of this work is to improve the resolution of the structural boundaries and decrease the percentage of transitional regions with the invariant design variable. Several 2D and 3D numerical examples indicate the effectiveness of our proposed method. Seen from the examples, the structural boundary become smoother and the intermediate densities have been reduced up to 70%. In addition, a design process based on the presented method is proposed to make the optimum solutions be fabricated conveniently and accurately by linking it with the 3D design software, ie, SolidWorks, which is also demonstrated in the numerical examples.

Topology optimization of continuum structures considering damage based on independent continuous mapping method

Continuum topology optimization usually produces results similar to a skeleton structure. In addition, material utilization in the optimized structure is greatly improved compared with the original structure. On the other hand, the redundancy of the structure is greatly reduced due to the removed material. Partial local failure in the optimized structure makes it more difficult for the entire structure to meet strength/stiffness requirements. Using the independent continuous mapping (ICM) method, with minimal weight as the objective and both stress and displacement as the respective constraints, continuum topology optimization models which also consider damage can be employed. A dual-sequence quadratic programming (DSQP) algorithm was used in this work to solve such topology optimization models. Numerical examples confirmed the effectiveness and feasibility of the models. The results indicated that both good load path and weight reduction can be obtained. In addition, compared with the structure obtained using conventional topology optimization, redundancy is greatly improved, and the strength/stiffness requirements for the structure can be satisfied for each damage scenario. Furthermore, the results indicate that the strength/stiffness of the structure, after topology optimization, is only slightly sensitive to local damage.