Note on spatial gradient operators and gradient-based minimum length constraints in SIMP topology optimization

2019 ◽  
Vol 60 (1) ◽  
pp. 393-400 ◽  
Author(s):  
Kaike Yang ◽  
Eduardo Fernandez ◽  
Cao Niu ◽  
Pierre Duysinx ◽  
Jihong Zhu ◽  
...  
Keyword(s):  
Topology Optimization ◽  
Minimum Length ◽  
Spatial Gradient ◽  
Gradient Based

A Density Gradient Approach to Topology Optimization Under Design-Dependent Boundary Loading

2018 ◽  
Author(s):  
Cunfu Wang ◽  
Xiaoping Qian
Keyword(s):  
Topology Optimization ◽  
Density Gradient ◽  
Optimization Method ◽  
Spatial Gradient ◽  
Computationally Efficient ◽  
Physical Density ◽  
Gradient Based ◽  
Design Dependent Loads ◽  
Previous Iteration ◽  
Gradient Approach

The paper proposes a density gradient based approach to topology optimization under design-dependent boundary loading. In the density-based topology optimization method, we impose the design dependent loads through spatial gradient of the density. We transform design-dependent boundary loads into a volume form through volume integral of density gradient. In many applications where loadings only need to be exerted on partial boundary, we introduce an auxiliary loading density to keep track of the loading boundary. During the optimization, the loading density is updated by tracking the changes of the physical density in the vicinity of the loading boundary at previous iteration. The proposed approach is easy to implement and computationally efficient. In addition, by adding more auxiliary density fields, the proposed approach is applicable to multiple design-dependent loads. To prevent the intersection of different loading boundaries, a Heaviside projection based integral constraint is developed. Both heat conduction problems under convection loading and elastic problems under hydrostatic pressure loading are presented to illustrate the effectiveness and efficiency of the method.

Gradient-Based Multi-Component Topology Optimization for Additive Manufacturing (MTO-A)

2017 ◽  
Author(s):  
Yuqing Zhou ◽  
Kazuhiro Saitou
Keyword(s):  
Topology Optimization ◽  
Additive Manufacturing ◽  
Simultaneous Optimization ◽  
Minimum Length ◽  
Manufacturing Constraints ◽  
Component Level ◽  
Component Size ◽  
Design Variables ◽  
Gradient Based ◽  
Minimum Length Scale

Topology optimization for additive manufacturing has been limited to the component-level designs with the component size smaller than the printer’s build volume. To enable the design of structures larger than the printer’s build volume, this paper presents a gradient-based multi-component topology optimization framework for structures assembled from components built by additive manufacturing. Constraints on component geometry for additive manufacturing are incorporated in the density-based topology optimization, with additional design variables specifying fractional component membership. For each component, constraints on build size, enclosed voids, overhangs, and the minimum length scale are imposed during the simultaneous optimization of overall base topology and component partitioning. The preliminary result on a minimum compliance structure shows promising advantages over the conventional monolithic topology optimization. Manufacturing constraints previously applied to monolithic topology optimization gain new interpretations when applied to multi-component assemblies, which can unlock richer design space for topology exploration.

Finding Better Local Optima in Topology Optimization via Tunneling

2018 ◽  
Author(s):  
Shanglong Zhang ◽  
Julián A. Norato
Keyword(s):  
Topology Optimization ◽  
Optimization Problems ◽  
Structural Response ◽  
Free Form ◽  
Local Optima ◽  
Design Representation ◽  
Tunneling Method ◽  
Optimization Parameters ◽  
Gradient Based ◽  
Geometric Primitives

Topology optimization problems are typically non-convex, and as such, multiple local minima exist. Depending on the initial design, the type of optimization algorithm and the optimization parameters, gradient-based optimizers converge to one of those minima. Unfortunately, these minima can be highly suboptimal, particularly when the structural response is very non-linear or when multiple constraints are present. This issue is more pronounced in the topology optimization of geometric primitives, because the design representation is more compact and restricted than in free-form topology optimization. In this paper, we investigate the use of tunneling in topology optimization to move from a poor local minimum to a better one. The tunneling method used in this work is a gradient-based deterministic method that finds a better minimum than the previous one in a sequential manner. We demonstrate this approach via numerical examples and show that the coupling of the tunneling method with topology optimization leads to better designs.

scholarly journals Data-Driven Additive Manufacturing Constraints for Topology Optimization

2018 ◽  
Author(s):  
Benjamin M. Weiss ◽  
Joshua M. Hamel ◽  
Mark A. Ganter ◽  
Duane W. Storti
Keyword(s):  
Topology Optimization ◽  
Additive Manufacturing ◽  
Optimal Designs ◽  
Data Driven ◽  
Constraint Function ◽  
Manufacturing Constraint ◽  
Gradient Based ◽  
Fixed Radius ◽  
Different Shapes ◽  
Moving Morphable Components

The topology optimization (TO) of structures to be produced using additive manufacturing (AM) is explored using a data-driven constraint function that predicts the minimum producible size of small features in different shapes and orientations. This shape- and orientation-dependent manufacturing constraint, derived from experimental data, is implemented within a TO framework using a modified version of the Moving Morphable Components (MMC) approach. Because the analytic constraint function is fully differentiable, gradient-based optimization can be used. The MMC approach is extended in this work to include a “bootstrapping” step, which provides initial component layouts to the MMC algorithm based on intermediate Solid Isotropic Material with Penalization (SIMP) topology optimization results. This “bootstrapping” approach improves convergence compared to reference MMC implementations. Results from two compliance design optimization example problems demonstrate the successful integration of the manufacturability constraint in the MMC approach, and the optimal designs produced show minor changes in topology and shape compared to designs produced using fixed-radius filters in the traditional SIMP approach. The use of this data-driven manufacturability constraint makes it possible to take better advantage of the achievable complexity in additive manufacturing processes, while resulting in typical penalties to the design objective function of around only 2% when compared to the unconstrained case.

Multi-Component Topology Optimization for Powder Bed Additive Manufacturing (MTO-A)

2018 ◽  
Author(s):  
Yuqing Zhou ◽  
Tsuyoshi Nomura ◽  
Kazuhiro Saitou
Keyword(s):  
Topology Optimization ◽  
Additive Manufacturing ◽  
Simultaneous Optimization ◽  
Previous Attempt ◽  
Numerical Instability ◽  
Feature Size ◽  
Powder Bed ◽  
Nonlinear Projection ◽  
Gradient Based ◽  
Length Width

This paper presents a gradient-based multi-component topology optimization (MTO) method for structures assembled from components made by powder bed additive manufacturing. It is built upon our previous work on the continuously-relaxed MTO framework utilizing the concept of fractional component membership. The previous attempt on the integration of the relaxed MTO framework with additive manufacturing constraints, however, suffered from numerical instability for larger size problems, limiting its application to 2D low-resolution examples. To overcome this difficulty, this paper proposes an improved MTO formulation based on a design field regularization and a nonlinear projection of component membership variables, with a focus on powder bed additive manufacturing. For each component, constraints on the maximum allowable build volume (i.e., length, width, and height), the elimination of enclosed voids, and the minimum printable feature size are imposed during the simultaneous optimization of the overall base topology and component partitioning. The scalability of the new MTO formulation is demonstrated by a few 2D examples with much higher resolution than previously reported, and the first reported 3D example of MTO.

Spatial-Gradient-Based Digital Volume Correlation Technique for Internal Deformation Measurement

2011 ◽  
Vol 31 (6) ◽  
pp. 0612005 ◽  
Author(s):  
潘兵 Pan Bing ◽  
吴大方 Wu Dafang ◽  
谢惠民 Xie Huimin ◽  
胡振兴 Hu Zhenxing
Keyword(s):  
Deformation Measurement ◽  
Digital Volume Correlation ◽  
Correlation Technique ◽  
Spatial Gradient ◽  
Gradient Based ◽  
Internal Deformation

Non-Probabilistic Based Topology Optimization Under External Load Uncertainty With Eigenvalue-Superposition of Convex Models

2013 ◽  
Author(s):  
Xike Zhao ◽  
Hae Chang Gea ◽  
Wei Song
Keyword(s):  
Topology Optimization ◽  
External Load ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Optimization Method ◽  
Convex Model ◽  
Structure Response ◽  
Gradient Based ◽  
Bounded Convex ◽  
Topology Optimization Method

In this paper the Eigenvalue-Superposition of Convex Models (ESCM) based topology optimization method for solving topology optimization problems under external load uncertainties is presented. The load uncertainties are formulated using the non-probabilistic based unknown-but-bounded convex model. The sensitivities are derived and the problem is solved using gradient based algorithm. The proposed ESCM based method yields the material distribution which would optimize the worst structure response under the uncertain loads. Comparing to the deterministic based topology optimization formulation the ESCM based method provided more reasonable solutions when load uncertainties were involved. The simplicity, efficiency and versatility of the proposed ESCM based topology optimization method can be considered as a supplement to the sophisticated reliability based topology optimization methods.

Topology Optimization of Micromachined Structures With the Manufacturing Constraints of KOH Etching of (110) Silicon

2008 ◽  
Author(s):  
Sudarshan Hegde ◽  
G. K. Ananthasuresh
Keyword(s):  
Topology Optimization ◽  
Aspect Ratio ◽  
High Aspect Ratio ◽  
Cost Effective ◽  
Wet Etching ◽  
Minimum Length ◽  
The Cost ◽  
Design Parameterization ◽  
Koh Etching ◽  
Si Wafer

The focus of this paper is on designing useful compliant micro-mechanisms of high-aspect-ratio which can be microfabricated by the cost-effective wet etching of (110) orientation silicon (Si) wafers. Wet etching of (110) Si imposes constraints on the geometry of the realized mechanisms because it allows only etch-through in the form of slots parallel to the wafer’s flat with a certain minimum length. In this paper, we incorporate this constraint in the topology optimization and obtain compliant designs that meet the specifications on the desired motion for given input forces. Using this design technique and wet etching, we show that we can realize high-aspect-ratio compliant micro-mechanisms. For a (110) Si wafer of 250 μm thickness, the minimum length of the etch opening to get a slot is found to be 866 μm. The minimum achievable width of the slot is limited by the resolution of the lithography process and this can be a very small value. This is studied by conducting trials with different mask layouts on a (110) Si wafer. These constraints are taken care of by using a suitable design parameterization rather than by imposing the constraints explicitly. Topology optimization, as is well known, gives designs using only the essential design specifications. In this work, we show that our technique also gives manufacturable mechanism designs along with lithography mask layouts. Some designs obtained are transferred to lithography masks and mechanisms are fabricated on (110) Si wafers.

Casting and Milling Restrictions in Topology Optimization via Projection-Based Algorithms

2012 ◽  
Author(s):  
James K. Guest ◽  
Mu Zhu
Keyword(s):  
Topology Optimization ◽  
Continuation Methods ◽  
Minimum Length ◽  
Manufacturing Constraints ◽  
Element Space ◽  
Structure Topology ◽  
Design Variables ◽  
Unstructured Meshing ◽  
Independent Design ◽  
Minimum Length Scale

Projection-based algorithms are arising as a powerful tool for continuum topology optimization. They use independent design variables that are projected onto element space to create structure topology. The projection functions are designed so that geometric properties, such as the minimum length scale of features, are naturally achieved. They therefore offer an efficient means for imposing geometry-related design specifications and/or manufacturing constraints. This paper presents recent advances in projection-based algorithms, including topology optimization under manufacturing constraints related to milling and casting processes. The new advancements leverage the logic of recently proposed algorithms for Heaviside projection, including eliminating continuation methods on projection parameters and potential for using multiple design variables to achieve active projection of each phase used in design. The primary advantages of such an approach are that manufacturing restrictions are achieved naturally, without need for additional constraints, and that sensitivity calculations are efficient and straightforward. The primary drawback of the approach is that the so-called neighborhood maps require storage for efficient processing when using unstructured meshing.

Sign in / Sign up

Export Citation Format

Share Document