Heaviside Projection Research Articles

Conductive paths generation using topology optimization for worst-case thermal design in space systems

Designing thermal systems for space platforms is a challenging task due to the wide variability in thermal conditions during the orbit and the strict constraints imposed by other subsystems. Several strategies have been developed to address these challenges while minimizing the thermal control subsystem’s signature, encompassing different algorithms optimize components positions in the platform. When relocation is not possible, thermal couplings between components can be modified to enhance heat transfer and achieve more uniform temperature distribution. To design optimal thermal paths in 2D space structures, in this paper a topology optimization framework based on the Heaviside Projection Method is introduced, using the Lumped Parameter Method (LPM) and taking the radiation heat transfer into account. The algorithm’s performance is tested on a Printed Circuit Board (PCB) by designing an optimal copper layer to meet specific thermal requirements under both extreme hot and cold conditions. Results show a significant improvement in thermal compliance for all components with the addition of this high-conductivity layer. Additionally, a reduction method is introduced to transfer the material distribution from the optimization to a coarser mesh of any size, which is useful to facilitate its implementation in existing software, where a large number of degrees of freedom can be a limitation.

An efficient volume-preserving binary filter for additive manufacturing in topology optimization

Topology optimization faces challenges concerning grey transition regions, manufacturability and computational efficiency. This article introduces an approach called the volume-preserving binary (VPB) filter. This novel sensitivity/density filter, based on a determinant-based matrix approach, efficiently detects grey elements and produces completely binary designs. The VPB filter addresses volume preservation by accurately expressing the volume-preserving condition, i.e. the volume before filtering equals the volume of solid materials after filtering. Comparison with other filtering methods, including the volume-preserving Heaviside filter and the Heaviside projection, illustrates the superior clarity and speed of the VPB filter. It reduces the number of iterations required for topology optimization, from hundreds or thousands to only tens of iterations, making it a promising solution for practical applications where manufacturability and computational efficiency are principal. Examples on compliance optimization, compliant mechanisms and heat conduction demonstrate the VPB filter's effectiveness in providing satisfactory solutions while perfectly preserving the length scale.

Topology Optimization Method of Stamping Structures Based on the Directional Density Field.

The stamping process produces thin-walled structures that, in general, have uniform wall thickness and no enclosed cavity. However, it is difficult to satisfy the above geometric requirements with the current density-based topology optimization method, since configuring the related geometric constraints is challenging. In order to solve this problem, a topology optimization method for stamping structures based on a directional density field is proposed. Specifically, the directional density field is developed to enable the adding and removing of materials only along the stamping direction, so as to avoid internal voids and concave features. The geometric control for uniform wall thickness is realized by tuning the truncation threshold of the Heaviside projection that processes the directional density field into the 0-1 binary field. At the same time, a calibrated filter radius of the truncation thresholds will facilitate the drawing angle control of the stamping ribs. The effectiveness of the established method has been verified by a number of numerical case studies. Results show that the proposed method can perform topology optimization for stamping structures with tunable uniform thickness and drawing angle control of the ribs. No internal voids or undercuts appear in the results. The results also disclose that a constant truncation threshold increment does not guarantee uniform wall thickness, and varying the threshold increments through surface offset and polynomial fitting is necessary.

Nonlinear topology optimization on thin shells using a reduced-order elastic shell model

We present a novel numerical algorithm to perform nonlinear topology optimization on elastic thin shells. The main component of our method is a differentiable thin-shell simulator based on discrete differential geometry (DDG) discretization and the projected Newton method to solve geometrically nonlinear elasticity and its derivatives on a triangle mesh. We build a density-based topology optimization algorithm, enhanced by a density filter and a Heaviside projection scheme, to emerge and optimize topologically complex shell structures on curved surfaces. We validate our approach using standard test cases for nonlinear topology optimization and demonstrate the efficacy of our method by tackling highly nonlinear topology optimization problems by producing complex and high-resolution shell structural designs under various load conditions.

Displacement-based structural identification using differentiable physics

Infrastructure is at constant risk of failure due to internal or hidden damage resulting from environmental conditions, variability in building materials, and fabrication errors. These risks can be mitigated by structural health monitoring, but it is difficult to monitor the state of the entire structure using discrete measurements. In this study, we propose a scheme for estimating spatially varying elastic modulus fields of a structural element using differentiable physics to support structural health monitoring efforts. Our approach is evaluated on two numerical model systems, a cantilever beam and a simply supported beam, and systematically evaluated for its sensitivity to initialization, boundary conditions, and a wide variety of spatially varying modulus fields. The results demonstrate the effectiveness of this approach in finding structures with matching displacement responses, while also highlighting its limitations in recovering accurate material properties, as is common in with all inverse methods. To address these limitations, we employ regularization techniques including blur filters and Heaviside projection to mitigate degeneracy and recover more complex fields. This work lays the foundation for more informed decision-making in structural health monitoring, with especially close applications to digital image correlation based approaches.

Multi‐material topology optimization method for transient thermal structure with constraint on permissible temperatures

AbstractExcessive temperature may lead material to performance degradation, ablation, and phase change. Thus, controlling the maximum temperature of used material within a safe range is of great significance to the safe operation of thermal structures. In this article, a density‐based multi‐material topology optimization method for transient thermal structures is proposed, in which the maximal temperature for each material is controlled not greater than its permissible temperature. In this method, a projection algorithm based on the PDE (partial differential equation) filter and Heaviside projection is set up and used to identify the domains occupied by each material. The maximal temperature of each specific material occupying domain is approximately expressed by the condensed integrals in the time and space domains. The permissible temperature constraint is satisfied by controlling the condensed integrals not great than a specific threshold value (depending on the corresponding material's permissible temperature). Several examples illustrate that the permissible temperature plays an important role in the selection and the distribution scheme of the candidate materials for the design of multi‐material thermal structures. The proposed method can effectively control the maximal temperature of each material within its constraint range, and achieve the rational design for multi‐material thermal structures. In addition, different thermal loading time may also lead to different material selection of structure. This work has potential applications in the lightweight design of thermal insulation structures.

Fail-safe topology optimization for multiscale structures

This paper presents a novel fail-safe topology optimization method for multiscale structures. The partial damage of both macroscopic and microscopic scales is considered for structural design. To ensure precision, the effective elasticity tensor obtained by the homogenization method is fitted as a high-order polynomial function. Meanwhile, the simplified models of partially damaged truss-like microstructure are employed to reduce the computational cost and the difficulty of fitting. Moreover, Heaviside projection is applied to speed up the convergence and yield a relatively clear configuration. Three numerical examples are tested to demonstrate that the optimized multiscale structures successfully obtain comprehensive performances than optimized solid structures when appropriate microstructure configurations are chosen. Besides, multiscale structures are more self-supporting than solid structures and thus more suitable for additive manufacturing due to the large number of gray elements diffused.

A general differentiable layout optimization framework for heat transfer problems

In the present work, we intend to demonstrate how to efficiently perform heat source layout design in a gradient-based way to maximally enhance the performance of heat transfer systems. To this end, a general and differentiable heat source layout optimization framework based on parameterized level set functions is proposed to optimize a thermal objective. Heaviside projection is utilized to realize an analytical description of the heat source intensity function, thus engaging a differentiable finite element analysis (FEA) implementation. The automatic differentiation technique is incorporated into this framework for sensitivity analysis, which can avoid tedious manual derivation work. Furthermore, gradient oscillations caused by improper finite element discretization are observed within components moving boundaries, thus possibly resulting in optimization failures. To remedy this problem, an adaptive multiresolution FEA method is proposed to enhance the awareness of component boundaries and eliminate this instability without introducing an extra FEA cost. Several numerical experiments are presented to illustrate positive effects of the adaptive multiresolution strategy and demonstrate the effectiveness of the proposed general and differentiable approach in heat conduction problems. To the best of our knowledge, it is the first time to realize such a framework for heat transfer systems with arbitrary-shaped heat sources. It is very promising to extend this framework to other physical field problems, or combine this framework with a deep learning-based surrogate in a unified way and implement a more efficient low-cost layout design in future research.

An Effective Topological Representation and Dimensionality Reduction Approach for Multi-Material Structural Topology Optimization

Abstract Topology optimization is among the most effective tools for innovative and lightweight structural designs. Multi-material design is considered to achieve better structural performance than single-material design. To significantly reduce the design space dimensionality and facilitate the optimization of multi-material structural design problems, this study proposes an effective topological representation and dimensionality reduction approach based on the material-field series expansion (MFSE) model. In the proposed method, a specified number of material phases is described within a single material field with a piecewise Heaviside projection function. The topology optimization problem is solved by determining the optimal MFSE coefficients. Owing to the single material-field topological description and series expansion, the number of design variables is independent of the finite element mesh and the number of material phases. In terms of dimensionality reduction, the proposed method outperformed all reported state-of-the-art algorithms for multi-material topology optimization. The validity and universality of the proposed method are illustrated in two- and three-dimensional numerical examples.

Topology optimization of scale-dependent non-local plates

The main objective of this work is to extend finite element-based topology optimization problem to the two-dimensional, size-dependent structures described using weakly non-local Cosserat (micropolar) and strongly non-local Eringen’s theories, the latter of which finds an application for the first time, to the best of Authors’ knowledge. The optimum material layouts that minimize the structural compliance are attained by means of Solid Isotropic Material with Penalization approach, while the desired smooth, mesh-independent, binary solutions are obtained using density filter accompanied by volume preserving Heaviside projection method. The algorithms are enhanced by including an element removal and reintroduction strategy to reduce the computational cost, and to prevent spurious excessive distortion of elements with very low density. Example problems of practical importance are investigated under the assumption of linear elasticity to validate the code and to clearly demonstrate the influence of internal length scales and different non-locality mechanisms on final configurations. Obtained macro-scale optimum topologies admit the characteristics of corresponding continuum theories, and appear to be in agreement with the mechanical response governed by particle interactions in micro/nanoscale.

Three-dimensional topology optimization considering overhang constraints with B-spline parameterization

In this paper, a B-spline parameterization based topology optimization method is presented to design 3D self-supporting structures for additive manufacturing (AM). B-spline parameterization being independent of the FE mesh is used to represent the density field characterizing the material layout of the structure. Heaviside projection and Boolean operations are applied to ensure the obtention of a clear topology configuration within a free-form design domain. Three specific constraints are formulated for AM: (i). The overhang angle constraint imposed on the structural boundaries in an integral form. (ii). The pyramid constraint aiming to avoid undesired inverted-cone features by constraining the values of density field at detection points uniformly distributed in the B-spline parameterized domain. (iii). The boundary gradient constraint imposed to deal with V-shaped cross-section features appearing on the boundary of the design domain by constraining the corresponding direction of the gradient of the B-spline density field. Meanwhile, the minimum feature size control is employed to deal with V-shaped cross-section features inside the design domain. Representative examples are dealt with to demonstrate the effectiveness of the proposed method. Numerical results show that the proposed method is effective to solve 3D problems.

HoneyTop90: A 90-line MATLAB code for topology optimization using honeycomb tessellation

This paper provides a simple, compact and efficient 90-line pedagogical MATLAB code for topology optimization using hexagonal elements (honeycomb tessellation). Hexagonal elements provide nonsingular connectivity between two juxtaposed elements and, thus, subdue checkerboard patterns and point connections inherently from the optimized designs. A novel approach to generate honeycomb tessellation is proposed. The element connectivity matrix and corresponding nodal coordinates array are determined in 5 (7) and 4 (6) lines, respectively. Two additional lines for the meshgrid generation are required for an even number of elements in the vertical direction. The code takes a fraction of a second to generate meshgrid information for the millions of hexagonal elements. Wachspress shape functions are employed for the finite element analysis, and compliance minimization is performed using the optimality criteria method. The provided Matlab code and its extensions are explained in detail. Options to run the optimization with and without filtering techniques are provided. Steps to include different boundary conditions, multiple load cases, active and passive regions, and a Heaviside projection filter are also discussed. The code is provided in Appendix~A, and it can also be downloaded along with supplementary materials from \url{https://github.com/PrabhatIn/HoneyTop90}.

Stress-constrained thermo-elastic topology optimization of axisymmetric disks considering temperature-dependent material properties

The lightweight design of turbine disks is challenging due to the extreme thermomechanical service environment which leads to high stress and temperature-dependency of material properties. The simplified engineering design method which uses uniform material properties at fixed temperature would lead to either unsafe or conservative designs. In this article, a stress-constrained thermo-elastic topology optimization method is proposed for axisymmetric disks considering temperature-dependent material properties. The temperature-dependent property functions fitted from the material test data are used to determine the material properties of each element according to the element’s average temperature. The strain energy constraint is added into the optimization formulation in order to prevent isolated material and severe violation of stress constraint caused by abrupt volume change. The Helmholtz filter and Heaviside projection are utilized to eliminate the checkerboard pattern and ensure distinct solid/void interfaces in the optimization results, respectively. The geometry extraction from the optimization results is easily implemented by the Least Square Fitting with B-spline Method. In the illustrative examples, the effectiveness of the proposed method in obtaining lighter disk designs is demonstrated in comparison to the design obtained by the simplified engineering method and the traditional single-web disk design. The ability of the proposed method to deal with arbitrary nonuniform temperature field is also demonstrated and it is learned that lighter disk design can be obtained if the temperature field has a lower radial temperature gradient at the disk hub.

Improved Efficient Projection Density Function Based on Topology Optimization

Topology optimization is a powerful tool having capability of generating new solution to engineering design problems, while these designs enhance manufacturability and reduce manufacturing costs in a computational setting. Mesh-independent convergence and other techniques have been widely used as topology optimization technique, but they produce gray transition regions which is not a favorable condition for any material. In this article, a modified topology optimization formulation using a new function has been proposed. The suggested scheme makes use of the Heaviside Projection Method (HPM) to continuum topology optimization. Such technique is helpful to obtain the minimum length scale influence on void and solid phases. Application of this proposed approach is implemented to obtain the minimum compliance for macrostructures. Numerical remarkable examples illustrate the noteworthy value of the proposed approach.

On the preliminary shape design of axisymmetric twin-web turbine discs considering the burst speed constraint

In this article, a topology optimization method is proposed to obtain a new preliminary shape design of the twin-web turbine disc considering the burst speed constraint. The burst speed constraint is converted to the constraints on the averaged hoop stress and the maximum averaged radial stress. The linear density filter and Heaviside projection are employed to prevent the mesh dependency and chequerboard pattern and ensure distinct solid/void interfaces in the optimization result. The geometric extraction is easily implemented using the least square fitting with B-spline method. Through illustrative examples, the importance of considering the burst speed constraint in the design of turbine disc is verified, which contributes towards obtaining a safer design at the preliminary design stage. The effectiveness of the proposed method in obtaining a lighter preliminary twin-web turbine disc design is also demonstrated compared with the parametric optimization method.

Analytical relationships for imposing minimum length scale in the robust topology optimization formulation

When using the robust topology optimization formulation in the density framework, the minimum size of the solid and void phases must be imposed implicitly through the parameters that define the density filter and the smoothed Heaviside projection. Finding these parameters can be time consuming and cumbersome, hindering a general code implementation of the robust formulation. Motivated by this issue, in this article, we provide analytical expressions that explicitly relate the minimum length scale and the parameters that define it. The expressions are validated on a density-based framework. To facilitate the reproduction of results, MATLAB codes are provided.

Stress-constrained topology optimization for material extrusion polymer additive manufacturing

Abstract This paper presents a comprehensive numerical and experimental study on stress-constrained topology optimization for Fused Deposition Modeling (FDM) additive manufacturing. The qp method is employed to avoid the singularity issue of stress-constrained problems. The P-norm function with stability transformation is adopted to build the global stress constraint with iterative corrections to eliminate the gap between the maximum local stress and the P-norm stress. The Heaviside projection is employed to generate clear-cut 0–1 designs. Two benchmark examples have been studied with the numerical algorithm. Experiments are performed on the topologically optimized MBB beam to investigate the impact of the FDM process parameters, including deposition path direction, building direction, and slicing layer height, on the resulted structural strength. The stress-constrained designs without and with Heaviside projection are comparatively tested with experiments. The stress-minimization designs subject to different P-norm parameters are compared both numerically and experimentally. Experiments show that the deposition path direction and the building direction evidently affect the derived structural strength. Moreover, overthin structural members may severely degrade the structural strength due to manufacturing and loading uncertainties.

Concurrent design of the free damping structure for minimizing the frequency response in a broad frequency band

This article develops a multiscale topology optimization model for the free damping structure to optimize the microstructure of the damping material and its macro layout. The objective is to minimize structural response in a broad frequency gap subject to volume constraints on both macro and micro levels. To overcome the convergence difficulty caused by the sharp jumping of the harmonic response around structural eigenfrequencies and maintain high-precision integral, the Gauss Legendre integration method is adopted to calculate the integration of objective function. Based on the density-based method, the sensitivity analysis of the objective function and the constraint is deduced. The mode acceleration method is used to reduce computational cost and a modified threshold Heaviside projection method is used to obtain the clear black-and-white designs. Several numerical examples are investigated to demonstrate the validity of the proposed model.

A multi-scale discrete material optimization model for optimization of structural topology and material orientations to minimize dynamic compliance

For fiber-reinforced composites, the combination of structural topology optimization and fiber angle design is an excellent way to suppress structural vibration more efficiently and acquire a lighter structure. This paper proposes a concurrent optimization model for fiber-reinforced composite structures, where the mathematical model is built by combining the polynomial interpolation scheme (PIS) and the Heaviside penalization of discrete material optimization (HPDMO). Therein, the former is adopted to avoid the localized mode phenomena due to the mismatch between element stiffness and mass. And the latter is used for multi-candidate fiber orientation design. The objective function is to minimize the integration of the dynamic compliance in a given frequency band subject to macro-volume constraint. In order to ease the heavy computational burden caused by hundreds of frequency steps in each iteration, too many freedom degrees, and discrete design variables in HPDMO, the modal acceleration method is used to obtain a high-precision estimation of displacement response, and a decoupled method suitable for multiphase fiber material sensitivity analysis is also developed. Finally, a modified threshold Heaviside projection is used to obtain a black-and-white design. Several numerical examples are presented to verify the validity and robustness of the proposed model.

Revisiting element removal for density-based structural topology optimization with reintroduction by Heaviside projection

We present a strategy grounded in the element removal idea of Bruns and Tortorelli (2003) and aimed at reducing computational cost and circumventing potential numerical instabilities of density-based topology optimization. The design variables and the relative densities are both represented on a fixed, uniform finite element grid, and linked through filtering and Heaviside projection. The regions in the analysis domain where the relative density is below a specified threshold are removed from the forward analysis and replaced by nodal boundary conditions. This brings a progressive cut of the computational cost as the optimization proceeds and helps to mitigate numerical instabilities associated with low-density regions. Removed regions can be readily reintroduced since all the design variables remain active and are modeled in the formal sensitivity analysis. A key feature of the proposed approach is that the Heaviside projection promotes material reintroduction along the structural boundaries by amplifying the magnitude of the sensitivities inside the filter reach. Several 2D and 3D structural topology optimization examples are presented, including linear and nonlinear compliance minimization, the design of a force inverter, and frequency and buckling load maximization. The approach is shown to be effective at producing optimized designs equivalent or nearly equivalent to those obtained without the element removal, while providing remarkable computational savings.