•A novel framework of topology optimization with overhang constraint is developed.•Overhang angle is estimated by linear fitting to the local element densities.•Hanging features are avoided by ...controlling minimum size of structural members.•A printable minimum compliance design with lateral member size control is obtained.
Structural topology optimization (Bendsøe and Kikuchi, 1988; Bendsøe and Sigmund, 2003; Deaton and Grandhi, 2014; Cheng and Olhoff, 1981; Xie and Steven, 1993) 1–5 provides a numerical tool for structural design with optimum performance. However, these structures could be too complex to be fabricated. Additive manufacturing (AM) enables the fabrication of these complex structures and is perfectly suitable for realizing the full potential of TO. However, AM has its manufacturing constraints too. The overhang constraint is one of these constraints. Components with small overhang angles or hanging features may deform, droop or warp, when fabricated using laser or electron beams in a layer-wise manner. This paper proposes a new approach to obtain optimum structural topology with consideration of the overhang constraint. We develop an effective method to estimate structural boundary normals of the optimum and intermediate designs with zigzag and blurry boundaries in SIMP by fitting local element density distribution with linear surfaces. By controlling the horizontal length of structural component, the hanging feature and too thin component are effectively suppressed. The element-wise overhang angle constraints and hanging feature constraints are aggregated as two single constraints on the volume fraction of the elements that violate these element-wise constraints. The structural topology optimization problem is solved by MMA. Numerical examples are given to demonstrate the effectiveness of the proposed algorithm.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
This study presents the incorporation of the effective gradient‐free proportional topology optimization algorithm into the framework of isogeometric analysis. The minimization of the compliance is ...considered, and the solid isotropic material with penalization method is used. The geometry, displacements, and density are all described by non‐uniform rational B‐spline (NURBS) basis functions. The density at an integration point is determined proportionally to its compliance. Then, the NURBS description of the density is constructed elementwise by deriving a relation between densities assigned to integration points and control points. The global NURBS description of the density for the whole domain is a blend of those from elements. Furthermore, a multiresolution scheme is presented by means of k$$ k $$‐refinement technique to enable the efficient performance for large‐scale problems. The accuracy and efficiency of the proposed approach are assessed through six numerical examples, including two‐ and three‐dimensional structures, with several rigorous tests and comparisons with the gradient‐based optimality criteria algorithm.
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SBCE, SBMB, UL, UM, UPUK
Among the various structural optimization tools, topology optimization is the widely used technique in obtaining the initial design of structural components. The resulting topologically optimal ...initial design will be the input for subsequent structural optimizations such as shape, size and layout optimizations. However, iterative solvers used in conventional topology optimization schemes are known to be computationally expensive, thus act as a bottleneck in the manufacturing process. In this paper, a novel deep learning‐based accelerated topology optimization technique with the ability to predict ductile material failure is presented. A Conditional Generative Adversarial Network (cGAN) coupled with a Convolutional Neural Network (CNN) is used to predict the optimal topology of a given structure subject to a set of input variables. Subsequently, the same cGAN is trained to predict the Von‐Mises stress contours on the optimal structure by means of color transformed image‐to‐image translations. The ductile failure criterion is evaluated by comparing the cGAN predicted maximum Von‐Mises stress with the yield strength of the material. The proposed novel numerical method is proven to arrive at the topologically optimal design, accompanying the material failure decision within a negligible amount of time but also maintaining a higher prediction accuracy.
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Summary
We present a methodical procedure for topology optimization under uncertainty with multiresolution finite element (FE) models. We use our framework in a bifidelity setting where a coarse and ...a fine mesh corresponding to low‐ and high‐resolution models are available. The inexpensive low‐resolution model is used to explore the parameter space and approximate the parameterized high‐resolution model and its sensitivity, where parameters are considered in both structural load and stiffness. We provide error bounds for bifidelity FE approximations and their sensitivities and conduct numerical studies to verify these theoretical estimates. We demonstrate our approach on benchmark compliance minimization problems, where we show significant reduction in computational cost for expensive problems such as topology optimization under manufacturing variability, reliability‐based topology optimization, and three‐dimensional topology optimization while generating almost identical designs to those obtained with a single‐resolution mesh. We also compute the parametric von Mises stress for the generated designs via our bifidelity FE approximation and compare them with standard Monte Carlo simulations. The implementation of our algorithm, which extends the well‐known 88‐line topology optimization code in MATLAB, is provided.
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Topology optimization is a technique that allows for increasingly efficient designs with minimal a priori decisions. Because of the complexity and intricacy of the solutions obtained, topology ...optimization was often constrained to research and theoretical studies. Additive manufacturing, a rapidly evolving field, fills the gap between topology optimization and application. Additive manufacturing has minimal limitations on the shape and complexity of the design, and is currently evolving towards new materials, higher precision and larger build sizes. Two topology optimization methods are addressed: the ground structure method and density-based topology optimization. The results obtained from these topology optimization methods require some degree of post-processing before they can be manufactured. A simple procedure is described by which output suitable for additive manufacturing can be generated. In this process, some inherent issues of the optimization technique may be magnified resulting in an unfeasible or bad product. In addition, this work aims to address some of these issues and propose methodologies by which they may be alleviated. The proposed framework has applications in a number of fields, with specific examples given from the fields of health, architecture and engineering. In addition, the generated output allows for simple communication, editing, and combination of the results into more complex designs. For the specific case of three-dimensional density-based topology optimization, a tool suitable for result inspection and generation of additive manufacturing output is also provided.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
•Analytical solution is given for optimum orientation of anisotropic microstructure.•An algorithm is developed for concurrent optimization of macro and microstructure.•The method is applicable to ...compliant mechanism and minimum compliance problems.
This paper presents a general formulation and solution method for the problem of two-scale concurrent topology optimization of cellular structure and its anisotropic materials. This formulation seeks simultaneous determination of the macro and micro structural topologies and the orientations of microstructures, such as a unit cell with anisotropic and inhomogeneous material properties. In the present formulation, the microstructure of cellular material is uniform in the entire macrostructure but with spatially-varying orientations. The solution method consists of two new features: (a) a mutual strain energy density-based approach proposed for analytically determining the orientations of fully anisotropic materials in compliant mechanism problems, which can also be modified to address minimum compliance designs and (b) an extended and fully-coupled moving iso-surface threshold (MIST) method and algorithm developed for solving the formulated concurrent optimization problem of cellular structures using two-scale physical response functions. The present formulation and algorithm are validated via studying numerical examples of concurrent optimum design of: (i) macro and micro topologies; (ii) macro topology and orthotropic material orientations; and (iii) macro and micro topologies and material orientations.
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•A Kriging-assisted multiscale method is proposed for maximizing natural frequency.•The shape interpolation method is adopted to gain sample microstructures.•Kriging to predict effective properties ...of spatially-varying microstructures.•The spatially-varying microstructures ensure a sufficiently large design space.
This paper proposes a Kriging-assisted multiscale topology optimization method for maximizing natural frequencies of inhomogeneous cellular structures, where spatially-varying microstructural configurations and their macroscopic distribution are simultaneously optimized. At the beginning, under macroscopic boundary conditions for a cellular structure, the configurations of multiple prototype microstructures are topologically optimized by the parametric level set method (PLSM) combined with the numerical homogenization approach. A kinematical connective constraint is considered to ensure the connectivity between adjacent prototype microstructures. Then, a shape interpolation method is adopted to interpolate shapes of the prototype microstructures, so as to generate a series of sample microstructures. Based on these samples, Kriging metamodels are constructed to predict the effective property of each microstructure within the macrostructure. Finally, the variable thickness sheet (VTS) method is applied to optimize the material distribution pattern at macroscale for maximizing the natural frequency of the cellular structure, where an efficient mode-tracking strategy based on modal assurance criterion (MAC) is employed to track the target mode accurately. Numerical examples are provided to test the performance of the proposed method in natural frequency optimization of cellular structures. The results indicate that the multiscale cellular structures obtained by the proposed method show higher natural frequency compared with the monoscale macrostructural and microstructural designs.
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A thermo‐elastic topology optimization with stress and temperature constraints is proposed to attack the complex multiphysics problem in this paper. Based on the rational approximation of material ...properties (RAMP), the coupled equations of mechanic and temperature field are solved. Two optimization problems, volume minimization with temperature and stress constraints, and traditional compliance minimization with volume and temperature constraints, are discussed for comparison. The stress stabilizing control scheme (SSCS) combined with global stress measure is presented to tackle highly nonlinear and local nature of stress with thermal expansion in varying temperature field. The adjoint method is applied to achieve the sensitivity of multiphysics field and the density function is updated utilizing the method of moving asymptotes (MMA). Representative examples are investigated to demonstrate the effectiveness and utility of the proposed method. Clear topology and stable iterative process can be obtained for complex coupled problem by means of the SSCS. Meanwhile, the topology with stress and temperature constraints has obvious sensitivity to even subtle change in temperature. The optimization design considering several stress constraints under multithermal conditions can work well in different temperature ranges.
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•A multiscale topology optimization method is proposed for realizing a thermal cloak.•A homogenization method is combined with level set-based topology optimization.•Objective functions are defined ...with a macroscopic temperature field.•Optimized cloaks composed of several microstructures show high cloaking functionality.•An additional objective function realizes the robustness of cloaking performance.
Artificially designed composite materials consist of microstructures, that exhibit various thermal properties depending on their shapes, such as anisotropic thermal conductivity. One of the representative applications of such composite materials for thermal control is the thermal cloak. This study proposed a topology optimization method based on a level set method for a heat conduction problem to optimally design composite materials that achieve a thermal cloak. The homogenization method was introduced to evaluate its effective thermal conductivity coefficient. Then, we formulated a multiscale topology optimization method for the composite materials in the framework of the homogenization method, where the microstructures were optimized to minimize objective functions defined using the macroscopic temperature field. We presented examples of optimal structures in a two-dimensional problem and discussed the validity of the obtained structures.
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This article presents the simultaneous magnetic and structural topology optimization of a transverse laminated synchronous reluctance machine rotor using a solid isotropic with material penalization ...(SIMP)-based approach with the globally convergent method of moving asymptotes (GCMMA) optimization algorithm. Magneto-structural single and multi-objective optimization formulations are presented. The single objective formulation maximizes the average torque subject to constraints on torque ripple, Von Mises stress, and compliance. The multiobjective formulation simultaneously maximizes average torque and minimizes compliance subject to constraints on torque ripple and Von Mises stress. In both optimization formulations, the electrical steel densities in the rotor mesh elements are the control variables. Including stress and compliance in the optimization formulation ensures a structurally feasible design. The design-dependent centripetal force contributed by each element in the rotor mesh is considered dependent on its density and mass. An intermediate density material issue is observed where the density of the material in a mesh element does not converge fully to air or electrical steel. This issue is addressed through the use of a thresholding mass function which forces the convergence of the mass but not the density. The two optimization formulations are compared for rotors operating at 4000 and 12 000 r/min. An initial magnetic only topology optimization is used to seed the magneto-structural topology optimization to decrease the computational time. The impact of the magnetic only seed and rotor mesh density on the optimization outcomes is also examined.
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