Simultaneous Topology Optimization Research Articles

Simultaneous optimization of shape, thickness and topology of shell structures using sigmoid function

Simultaneous optimization of shape, thickness and topology of shell structures using sigmoid function

Simultaneous shape and topology optimization of structures under pressure load

Simultaneous shape and topology optimization of structures under pressure load

Integrated component-support topology optimization for additive manufacturing with post-machining

PurposeThe purpose of this paper is to communicate a method to perform simultaneous topology optimization of component and support structures considering typical metal additive manufacturing (AM) restrictions and post-print machining requirements.Design/methodology/approachAn integrated topology optimization is proposed using two density fields: one describing the design and another defining the support layout. Using a simplified AM process model, critical overhang angle restrictions are imposed on the design. Through additional load cases and constraints, sufficient stiffness against subtractive machining loads is enforced. In addition, a way to handle non-design regions in an AM setting is introduced.FindingsThe proposed approach is found to be effective in producing printable optimized geometries with adequate stiffness against machining loads. It is shown that post-machining requirements can affect optimal support structure layout.Research limitations/implicationsThis study uses a simplified AM process model based on geometrical characteristics. A challenge remains to integrate more detailed physical AM process models to have direct control of stress, distortion and overheating.Practical implicationsThe presented method can accelerate and enhance the design of high performance parts for AM. The consideration of post-print aspects is expected to reduce the need for design adjustments after optimization.Originality/valueThe developed method is the first to combine AM printability and machining loads in a single topology optimization process. The formulation is general and can be applied to a wide range of performance and manufacturability requirements.

Topology Optimization of Structures Made of Discrete Geometric Components With Different Materials

We present a new method for the simultaneous topology optimization and material selection of structures made by the union of discrete geometric components, where each component is made of one of multiple available materials. Our approach is based on the geometry projection method, whereby an analytical description of the geometric components is smoothly mapped onto a density field on a fixed analysis grid. In addition to the parameters that dictate the dimensions, position, and orientation of the component, a size variable per available material is ascribed to each component. A size variable value of unity indicates that the component is made of the corresponding material. Moreover, all size variables can be zero, signifying the component is entirely removed from the design. We penalize intermediate values of the size variables via an aggregate constraint in the optimization. We also introduce a mutual material exclusion constraint that ensures that at most one material has a unity size variable in each geometric component. In addition to these constraints, we propose a novel aggregation scheme to perform the union of geometric components with dissimilar materials. These ingredients facilitate treatment of the multi-material case. Our formulation can be readily extended to any number of materials. We demonstrate our method with several numerical examples.

Simultaneous topology and size optimization of a 3D variable-length structure described by the ALE–ANCF

An efficient optimization approach is proposed to simultaneously optimize the topology and geometrical sizes of a three-dimensional (3D) variable-length structure. First of all, in a unified global coordinate system, a flexible multibody system (FMBS) model is established to describe the extension and contraction dynamics of a variable-length structure. The variable-length flexible structure is meshed by the proposed arbitrary Lagrangian–Eulerian (ALE) solid elements of absolute nodal coordinate formulation (ANCF). To consider the variable-length behaviors, the classic equivalent static loads (ESL) is redefined by introducing the concept of virtual design domain. With the help of the ESL, the dynamic response optimization of the variable-length structure is turned into a static one. A simultaneous topology and size optimization approach is initially proposed via the moving morphable components (MMC). By appropriately arranging the initial configuration of the components and linking all or part of the design variables of a component with others’, the variable-length structure can be designed as a homogeneous or heterogeneous periodic structure. The approach is numerically implemented via an additional constraint that modifies the sensitivities with respect to some design variables. Finally, three numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed optimization approach.

Simultaneous topology, shape, and size optimization of trusses, taking account of uncertainties using multi-objective evolutionary algorithms

This paper proposes the design of trusses using simultaneous topology, shape, and size design variables and reliability optimization. Objective functions consist of structural mass and reliability, while the probability of failure is set as a design constraint. Design variables are treated to simultaneously determine structural topology, shape, and sizes. Six test problems are posed and solved by a number of multi-objective evolutionary algorithms, and it is found that Hybridized Real-Code Population-Based Incremental Learning and Differential Evolution is the best performer. This work is considered an initial study for the combination of reliability optimization and simultaneous topology, shape, and sizing optimization of trusses.

Simultaneous optimization of the district heating network topology and the Organic Rankine Cycle sizing of a geothermal plant

This contribution presents the optimization of parallel distribution between electricity and heat production for a geothermal plant. The geothermal fluid is split into two streams, one used for an Organic Rankine Cycle (ORC) system, and the other for a District Heating Network (DHN). The superstructure to be used for the optimization problem includes the ORC components, one of which is an optional internal heat exchanger which allows exchange between the outlet streams of the turbine and the pump. Each of the components' characteristic dimensions (used in the installation cost) is an optimization variable. The operating cost of the ORC is proportional to the installation cost. The superstructure also includes the DHN topology constituted by a definite consumer and optional consumers. A Mixed Integer Non-Linear Programming (MINLP) optimization problem is formulated and solved using the GAMS software. The strategy used to overcome the critical point of the initialization of the MINLP problem is presented. It consists in dividing the general problem into sub-problems which are solved successively. Three different academic study cases are compared to a reference case. The results validate the stability and the robustness of this optimization tool. A sensitivity analysis is performed in geothermal source conditions. All these results highlight the relevance of the simultaneous approach.

Simultaneous optimization of topology, control and size for multi-mode hybrid tracked vehicles

Hybrid tracked vehicles have become increasingly popular for off-road applications due to their better fuel economy and higher output power. Currently, the most popular in the production of tracked vehicles are the series hybrid, because of the simple powertrain designs. However, they suffer from high energy conversion losses and large propulsion motors. To overcome these issues, multi-mode hybrid tracked vehicles are employed since they have high efficiency and excellent overall performance. The proposed multi-mode hybrid powertrain can realize straight driving, turning, and driving backwards without any additional steering mechanism. To systematically explore all the possible designs of multi-mode hybrid designs with planetary gears, a topology-control-size-integrated optimization approach is presented. A novel near-optimal energy management strategy, Efficiency Evaluation Real-time Control Strategy (EERCS), is proposed to rapidly calculate near-optimal control rules for design candidates. The EERCS is confirmed to achieve results similar to those of Dynamic Programming (DP), yet the computation time is over 50 times less. With the help of EERCS, the optimal design together with its parameters is computed using multi-objective optimization based on a meta-heuristic algorithm. Results of a case study show that the optimized design with downsized components produces improved drivability and fuel economy compared to the series hybrid benchmark.

Simultaneous shape and topology optimization method based on the H<sup>1</sup> gradient method for creating light weight plate and shell structures

In this paper, we present a simultaneous optimization method of shape and topology for designing a light-weight plate and shell structure. The free-form optimization method for shells and SIMP method are respectively employed for shape and topology optimization, and combined effectively. Shape and fictitious homogenized-density variations are used as the design variables, and simultaneously determined in one iteration of the convergence process. With this method, the optimal topology is determined in the variable design surface optimized by shape optimization. Compliance is used as the objective functional, and minimized under the volume constraint. The optimal design problem is formulated as a distributed-parameter optimization problem, and the sensitivity functions with respect to shape and density variations are theoretically derived. Both the optimal shape and density variations are determined by the H1 gradient method, where the sensitivity functions are applied as the Robin condition to the design surface. With the proposed method, the optimal lighter and stiffer shell structure with smooth surface can be obtained without any design parameterization and numerical instabilities such as checkerboard and zigzag-shape problems.

Gradient-based multi-component topology optimization for stamped sheet metal assemblies (MTO-S)

This paper presents a continuous relaxation of our previous work on the multi-component topology optimization, which enables the use of efficient gradient-based optimization to attain the simultaneous optimization of a base structural topology and its decomposition. In addition to the fictitious material density used in the classic SIMP method, new design variables specifying fractional membership to each component are introduced, in order to relax the process of decomposing a candidate structural topology into a discrete number of components. The concept of joint stiffness is also relaxed using the new component membership variables. Considering stamped sheet metal assemblies jointed by the resistance spot-welding, the stamping die cost consists of die-set material cost and die machining cost. The former is modeled as the minimum bounding box area enclosing each component, approximated by a product of weighted variances in major and minor directions obtained by a weighted principal component analysis. The latter is modeled as an empirical shape complexity index similar to the one available in the literature. Examples on compliance minimization of multi-component sheet metal assemblies are presented, where the proposed continuous relaxation formulation solved by a gradient-based optimization algorithm generated comparable results with dramatically improved computational efficiency over our previous results by discrete formulations solved by the genetic algorithm.

Simultaneous shape and topology optimization of prestressed concrete beams

This paper presents a new optimization approach for the design of prestressed concrete beams. The prestressing tendon is modeled as a chain of linear segments that transfer point forces to the concrete domain according to the tendon’s angles. The concrete beam is modeled as a discretized continuum following density-based approaches to topology optimization. The shape of the tendon and the topology of the surrounding concrete are optimized simultaneously within a single problem formulation. A special filtering technique is developed in order to ensure that the tendon is covered by concrete, thus shape and topological variables are tightly coupled. Several test cases demonstrate the applicability of the proposed optimization procedure. The deformation of the optimized designs due to external loads is counteracted by the deformation due to prestressing, hence by tuning the force in the tendon the total deformation can approach zero. Consequently, the beams exhibit a compression-only response meaning that the common goal of prestressed concrete design is achieved.

Simultaneous topology optimization of supporting structure and loci of isolators in an active vibration isolation system

We developed a new multi-objective and multi-level optimization method to design an active vibration isolation system. Both the layout of the continuum (i.e. the supporting structure) and the loci of the isolators are designed using topology optimization technique in a unified formulation for the first time. The static, dynamic and vibration-isolation characteristics are taken into account simultaneously in the present model. Due to their different roles in the system it may be appropriate and advantageous to treat the design of the continuum layout and isolator loci as different sub-problems with different objectives in separate stages. The multi-level optimization technique, where the optimization of the supporting structure and the isolator loci are incorporated into a closed-loop, is proposed and implemented so that the interactions between these two sub-problems can be fully taken into account. Numerical results demonstrate the validity of the proposed design cycle. Comparisons show that the overall static, dynamic and vibration-isolation performance of the optimized system outperforms the ones designed by traditional methods.

Size, shape, and topology optimization of planar and space trusses using mutation-based improved metaheuristics

Abstract In this study, simultaneous size, shape, and topology optimization of planar and space trusses are investigated. Moreover, the trusses are subjected to constraints for element stresses, nodal displacements, and kinematic stability conditions. Truss Topology Optimization (TTO) removes the superfluous elements and nodes from the ground structure. In this method, the difficulties arise due to unacceptable and singular topologies; therefore, the Grubler's criterion and the positive definiteness are used to handle such issue. Moreover, the TTO is challenging due to its search space, which is implicit, non-convex, non-linear, and often leading to divergence. Therefore, mutation-based metaheuristics are proposed to investigate them. This study compares the performance of four improved metaheuristics (viz. Improved Teaching–Learning-Based Optimization (ITLBO), Improved Heat Transfer Search (IHTS), Improved Water Wave Optimization (IWWO), and Improved Passing Vehicle Search (IPVS)) and four basic metaheuristics (viz. TLBO, HTS, WWO, and PVS) in order to solve structural optimization problems. Highlights Improvements in four recently designed metaheuristics. Use of random mutation-based search technique. Applications on challenging/benchmark problems of simultaneous size, shape, and topology optimization of truss structures. Improvements effective over basic metaheuristics.

New biomimetic approach to the aircraft wing structural design based on aeroelastic analysis

Abstract This paper presents a new biomimetic approach to the structural design. For the purpose of aircraft wing design the numerical environment combining simultaneous structural size, shape, and topology optimization based on aeroelastic analysis was developed. For the design of aircraft elements the optimization process must be treated as a multi-load case task, because during the fluid structure interaction analysis each step represents a different structural load case. Also, considering different angles of attack, during the CFD computation each result is considered. The method-specific features (such as domain independence, functional configurations during the process of optimization, and multiple load case solution implemented in the optimization scenario) enable the optimal structural form. To illustrate the algorithm functionality, the problem of determining the optimal internal wing structure was presented. The optimal internal wing structure resulting from aeroelastic computation with different angles of attack has been presented.

Concurrent multi-scale design optimization of composite frames with manufacturing constraints

This paper presents a gradient based concurrent multi-scale design optimization method for composite frames considering specific manufacturing constraints raised from the aerospace industrial requirements. Geometrical parameters of the frame components at the macro-structural scale and the discrete fiber winding angles at the micro-material scale are introduced as the independent design variables at the two geometrical scales. The DMO (Discrete Material Optimization) approach is utilized to couple the two geometrical scales and realize the simultaneous optimization of macroscopic topology and microscopic material selection. Six kinds of manufacturing constraints are explicitly included in the optimization model as series of linear inequalities or equalities. The capabilities of the proposed optimization model are demonstrated with the example of compliance minimization, subject to constraint on the composite volume. The linear constraints and optimization problems are solved by Sequential Linear Programming (SLP) optimization algorithm with move limit strategy. Numerical results show the potential of weight saving and structural robustness design with the proposed concurrent optimization model. The multi-scale optimization model, considering specific manufacturing constraints, provides new choices for the design of the composite frame structure in aerospace and other industries.

Sizing and topology optimization of truss structures using genetic programming

This paper presents a genetic programming approach for simultaneous optimization of sizing and topology of truss structures. It aims to find the optimal cross-sectional areas and connectivities of the joints to achieve minimum weight in the search space. The structural optimization problem is subjected to kinematic stability, maximum allowable stress and deflection. This approach uses the variable-length representation of potential solutions in the shape of computer programs and evolves to the optimum solution. This method has the capability to identify redundant truss elements and joints in the design space. The obtained results are compared with existing popular and competent techniques in literature and its competence as a tool in the optimization problem are demonstrated in solving some benchmark examples, the proposed approach provided lighter truss structures than the available solutions reported in the literature.

Force density method for simultaneous optimization of geometry and topology of trusses

A new method of simultaneous optimization of geometry and topology is presented for plane and spatial trusses. Compliance under single loading condition is minimized for specified structural volume. The difficulties due to existence of melting nodes are successfully avoided by considering force density, which is the ratio of axial force to the member length, as design variable. By using the fact that the optimal truss is statically determinate with the same absolute value of stress in existing members, the compliance and structural volume are expressed as explicit functions of force density only. After obtaining optimal cross-sectional area, nodal locations, and topology, the cross-sectional areas and nodal coordinates are further optimized using a conventional method of nonlinear programming. Accuracy of the optimal solution is verified through examples of plane trusses and a spatial truss. It is shown that various nearly optimal solutions can be found using the proposed method.

Simultaneous topology and size optimization of locomotive structure using multinary genetic algorithms

Structural optimization of a typical locomotive carbody subjected to various operational loads is investigated using a simultaneous topology and size optimization approach. The proposed approach builds on a string representation of the structure in which the presence or absence of a structural element between specific nodes, as well as its cross-section, are represented by integers. Strings representing various structure topologies are then evolved according to a modified Genetic algorithm with improved genetic operators. The efficiency of the proposed model is verified through applying it to optimize the carbody structure of ER24PC locomotive. Operational load cases and performance criteria are adopted from the European standard EN12663. It is shown that the results based on the present optimal design have higher safety factors compared to the original design without a significant increase in weight. Besides, the computational cost of the optimization process is shown to be considerably less than that of the classical binary Genetic algorithm.

Simultaneous optimization of shape, thickness and topology of shell structures using sigmoid function

Simultaneous optimization of shape, thickness and topology of shell structures using sigmoid function

Simultaneous optimization of Shape and Topology of laminated shell structures for minimizing multi-objective compliance

Simultaneous optimization of Shape and Topology of laminated shell structures for minimizing multi-objective compliance