Three-dimensional shape optimization using fully automatic mesh generation

CAD models serve as the fundamental carriers for representing geometric structures and design logic. Understanding the underlying construction logic of complex shapes facilitates the reuse, editing, and optimization of designs. Constructive Solid Geometry (CSG) offers a viable approach for shape parsing by representing complex geometries through combinations of simple primitives. This paper proposes a novel method for parsing 3D CAD shapes based on CSG representations. Leveraging quadric surfaces as the fundamental modeling primitives, our approach employs a dual-branch architecture— “composition–subtraction”—to construct target 3D shapes via Boolean operations. Each branch receives either voxel grids or point clouds as input and uses an auto-encoder to extract shape features, which are then used to predict a set of intermediate shapes. The final reconstruction is obtained through set difference operations. To enhance the model’s expressive capacity under a fixed Boolean operation sequence, we introduce rotation parameters and dual primitive designs. Moreover, we adopt a staged training framework that progressively guides the model from learning approximate signed distance functions to precise structural modeling. A regularization loss is incorporated to promote geometric diversity and plausibility of the primitives. After training, we further propose a post-processing strategy based on signed distance influence to remove redundant primitives, resulting in more compact and interpretable CSG sequences. Experimental comparisons on the ABC dataset demonstrate that our method reduces the 3D chamfer distance and edge chamfer distance by 72.1% and 31.1%, respectively, and improves the normal vector consistency by 9.48%, compared to SECAD-Net, and achieves better results in visual metrics.

Structural topology optimization using multi-objective genetic algorithm with constructive solid geometry representation

Robust event-driven particle tracking in complex geometries

Set operation evaluation is an important component of Geometric Modeling System (GMS). Input data can be a CSG (Constructive Solid Geometry) representation with solid primitives approximated by polyhedra. This representation is a powerful tool for solid object description. Output data will be the unique resulting polyhedral object which provides an efficient data structure in display field. With no use of spatial coherency, computational complexity of set operation is quadratic. In this paper, we are introducing a new space subdivision scheme called Boolean Octree which performs set operations on polyhedra in an efficient way. This structure aims at limiting set operation evaluation in a ‘minimal space of calculation’ where primitive boundaries intersect each other and where resulting evaluation participates in the construction of the final resulting object. Boolean Octree computes set operations in a local level providing a linear complexity for geometric calculations. During space subdivision, Boolean Octree has a global view on local CSG tree (projection of the CSG tree in local space) taking into account simplifications of the boolean expression, avoiding evaluation and subdivision for the object parts out of the ‘minimal space of calculation’.

Recently developed solid modelling systems for the design of complex physical solids using interactive computer graphics offer the exciting possibility of an integrated design/analysis system. Called geometric modellers, these systems build complex solids from primitive solids (cubes, cylinders, spheres, solid patches, etc.) and macro solids (combination of primitives)3, 4, 8, 16, 18, 25, 38. To provide an effective structural analysis capability for these systems, methods must be devised to ease the burden of discretizing the solid geometry into a user controlled (usually locally graded) finite element mesh. The purpose of this paper is to describe an interactive solid mesh generation system capable of generating valid meshes of well‐proportional tetrahedral finite elements for the decomposition of multiply connected solid structures. The system uses a semi‐automatic node insertion procedure to locate element node points within and on the surface of a structure. An independent automatic three‐dimensional triangulator then accepts these nodes as input and connects them to form a valid finite element mesh oftetrahedral elements. Although this report makes use of a modeller based on a constructive solid geometry representation (a so‐called CSG modeller), the mesh generation strategy elaborated herein is completely general and makes no particular use of the CSG representation.

A Cartesian grid generation method is developed in this study. Two kinds of solid modeling methods, CSG and STL models, are used for complicated solid modeling. The staircase boundary approximation is implemented to handle irregular geometries and the computational domain is discretized using a regular Cartesian grid. Using the edge-based integral slice algorithm, the models are sliced with a set of parallel planes to generate 2D slices information. The scan line filling technique is used to achieve grid generation after slicing. Two grid generation examples with a CSG model and a STL model are given to test the capability of the grid generation method. For grid displaying, a method is proposed to remove the hidden surfaces fasten based on the topology of orthogonal hexahedral grids. The parallelization of grid displaying is achieved by employing multi-threaded parallel technique. Parallel test results show that the parallel algorithm has the absolute advantage on speed compared to the serial algorithm.

This paper presents a unified differentiable boolean operator for implicit solid shape modeling using Constructive Solid Geometry (CSG). Traditional CSG relies on min, max operators to perform boolean operations on implicit shapes. But because these boolean operators are discontinuous and discrete in the choice of operations, this makes optimization over the CSG representation challenging. Drawing inspiration from fuzzy logic, we present a unified boolean operator that outputs a continuous function and is differentiable with respect to operator types. This enables optimization of both the primitives and the boolean operations employed in CSG with continuous optimization techniques, such as gradient descent. We further demonstrate that such a continuous boolean operator allows the modeling of both sharp mechanical objects and smooth organic shapes with the same framework. Our proposed boolean operator opens up new possibilities for future research toward fully continuous CSG optimization.

It is important to establish an integrated CAD system for mechanical engineering. The geometric model is constructed with primitives, such as wire frame, surface and solid on this system. The finite element model is induced from the substructure, which is a component of the geometric model. After the extraction of the substructure from the model, the mesh for the finite element analyses is automatically generated against the substructure. An approach to automatic mesh generation is discussed on this paper, based upon the techniques of interpolation with the blending function, originally developed for geometric modeling. But, the blending function may produce unexpected results when the mesh is generated, due to the nature of the algorithm. In order to conclude with expected results for automatic mesh generation, and accuracy-fitting method is introduced.

This paper deals with shape optimization of three-dimensional elastic bodies. The proposed approach is based on natural design variables and shape functions and utilizes the boundary element method. The design variables are the magnitudes of a set of fictitious loads applied on the structure. Shape optimal design problems based on minimum mean compliance, peak stress minimization and stress constraints are considered. A general method for shape design sensitivity analysis using the material derivative concept and adjoint variable method is used. The sensitivity formula for a general stress constraint imposed over the optimized boundary is derived. Optimal shapes for three-dimensional problems are presented to show numerical applications.

The first attempt to improve battery performance through model-based design optimization was made by W. Tiedemann and J. Newman in 19751. They varied the electrode thickness and porosity and then observed the effects on the effective capacity. Newman later applied the reaction-zone model to maximize the specific energy of the system using the same varying and observing method2. V. Ramadesigan et al.3developed optimal spatially varying porosity for porous electrodes by considering ohmic resistance model in the electrode. They minimized the ohmic resistance in the positive electrode by varying porosity using control vector parameterization (CVP) method. They also introduced the idea of using graded electrode to further improve the performace. Since the introduction of the pseudo-2D (P2D) model by the Newman group, many efforts have been made in utilizing the P2D model in battery design. W. Du et al. applied an optimization framework based on surrogate method4. S. Golmon et al. performed the multi-objective and multi-design-parameter optimization problem with adjoint sensitivity analysis5 incorporating mechanical stress-strain relationship. S. De and his coworkers6 used a reformulated model, to perform simultaneous optimization of multiple design parameters. N. Xue et al. 7 applied the gradient-based algorithm framework to optimize the cell design. More recently, Y. Dai and V. Srinivasan8used a gradient-free direct search method to maximize the specific energy. Z. Du et al. studies the effects of various design parameters on thick electrode performance9. Most of the aforementioned optimization research, used the sequential optimization approach, also known as CVP. CVP takes the differential algebraic equation (DAE) system describing the battery physics, applies a certain nonlinear programming (NLP) solver to them, discretizes only the control variables, and solves the model. The only exception being the effort by Golmon et al which used an indirect approach. Note that as of today, global optimization cannot be guaranteed when CVP type approaches are used with P2D type models. In general optimization approaches can be classified as indirect and direct approach. In the direct approach there are two main categories – sequential and simultaneous approach. The simultaneous approach10, on the other hand, discretizes both the control variables and the design variables before solving the problem. When used for optimization, the DAE system will only be solved once at the optimal point, compared to the repeated numerical integration needed for sequential optimization. By using high order discretization scheme on both the state and the design variables, simultaneous optimization approach results in faster determination of the optimum with fewer time steps. Furthermore, this approach offers more flexibility over constraints on the state variables. Since the state variables are also discretized, it is possible to apply equality/inequality constraints on their internal values. In this work, we will first apply the simultaneous optimization approach to a resistance model similar to the one mentioned earlier3, and compare its performance with the sequential approach. We will also explore the application of the same optimization approach to the P2D model, and identify the challenges. It is viewed as a controversial topic in the literature wherein different authors disagree on the relevance of graded porosity for improved performance8. By using a simultaneous optimization approach and guaranteeing global optimization for convex problem statements, we hope to provide additional insights. Acknowledgements The authors are thankful for the financial support by the Clean Energy Institute at the University of Washington and the Assistant Secretary for Energy Efficiency and Renewable Energy, Office of Vehicle Technologies of the U.S. Department of Energy through the Advanced Battery Materials Research (BMR) Program (Battery500 Consortium).

Effects of loading conditions and objective function on three-dimensional shape optimization of femoral components of hip endoprostheses

In this paper, our effort to apply 3-D Virtual Reality system for stereoscopic visualization of mesh data on the web is briefly described. This study is an extension of our previous and on-going research efforts to develop an automatic grid generation program specialized for wing mesh, named as eGWing. The program is developed by using JAVA programming language, and it can be used either as an application program on a local computer or as an applet in the network environment. In this research advancing layer method(ALM) augmented by elliptic smoothing method is used for the structured grid generation. And to achieve a stereoscopic viewing capability, two graphic windows are used to render its own viewing image for the left and right eye respectively. These two windows are merged into one image using 3D monitor and the viewers can see the mesh data visualization results with stereoscopic depth effects by using polarizing glasses. In this paper three dimensional mesh data visualization with stereoscopic technique combined with 3D monitor is demonstrated, and the current achievement would be a good start-up for further development of low-cost high-quality stereoscopic mesh data visualization system which can be shared by many users through the web.

A comprehensive grid generation system has been developed which enables grid generation for complex aircraft configurations. This system is especially suited for generation of zonal grids without point continuity at zoneto-zone interfaces. This approach reduces the complexity of the grid generation process by allowing grids to be generated with fewer zones, and fewer unnecessary grid points, thereby reducing solution time and cost. The capabilities of the system include geometry acquisition and manipulation, grid generation on the boundaries of each zone, three-dimensional field grid generation, grid quality evaluation for the field grid, boundary condition specification, coupling parameter identification for zone-to-zone interfaces, and output directly compatible with a zonal Euler or Navier-Stokes flowfield analysis code. The system operates in an interactive environment through a userfriendly graphical interface running on several computer platforms with numerous graphics devices. In order to further improve the efficiency of the grid generation process, the system utilizes several macro operations to perform sequences of steps ranging from generating various farfield boundary shapes to generating an entire zone for an isolated forebody or diffuser type geometry. A prototype expert system has also been implemented to assist the user in identifying and correcting flaws in the field grid. Applications of the system to inlet-forebody analyses, a duct with turning vanes, and a thrust-vectoring nozzle configuration will be presented. I n t r o d u c t i o n Numerical solution of the Navier-Stokes equations is becoming increasingly important for the analysis of advanced propulsion systems and their integration with an airframe. The ability to generate a suitable grid is often the pacing i tem in obtaining solutions for these complex configurations. High quality grids are necessary to accurately predict flowfields. High quality grids a180 increase the convergence rate of the solution, thereby re* Lend Engineer Aemdynsmics ** Technical Specisiixt Aerodyrmnics t Senior Engineer Aerodynamics t Technical Specialist Propulsion 5 Senior Engineer Propulsion Copyright @ 1991 by McDonnell Douglas Cormration Published by the American Institute 01 Aeronautics and Astronautics, Inc. with permission. ducing computer resource requirements. The grid must accurately represent the geometry. It must also be tnilored to the flowfield, Le., i t must provide high density near solid boundaries, shock locations, and other regions where high Bowfield gradients may be experienced. Any flaws in the grid must be identified and eliminated before the grid is used in the flowfield solution algorithm or else costly regridding and repeated analysis will be required. The McDonnell Aircraft Computational Grid System (MACGS) was developed to address these issues. It uses an interactive zonal approach which does not require point continuity a t zone-to-zone interfaces. The zonal approach maximizes flexibility and simplifies the grid gcneration problem. In a zonal approach, the physical domain around the configuration is subdivided into simpler regions 01 zones. These zones are non-overlapping and ., boundary conforming. A zonal approach dlows various strategies for subdividing the field into zones or the use of a single zone, where appropriate. Current zonal grid generation systems, such as EAGLE (Reference 1) and GRIDGEN (Reference Z), are highly focused on generation of grids with point continuity at the zonal interfaces. Although these grid generators are very versatile and allow zones without point continuity to be generated independently, these systems lose some of their power for such cases. They do not address how adjacent zones will be treated. MACGS generates grids without point continuity at the zonal interface and computes the interpolation factors t o allow flowfield da t a to be transferred between zones. Lack of point continuity restrictions makes grid generation much more flexible because the user can distribute points where they are needed, based on flow properties within the zone rather than on where points were distributed i n an adjacent zone. This allows the user t o have as many grid points as required in a critical flowfield region, such as a boundary layer, with far fewer points in a less critical adjacent zone, such as the region upstream of an inlet. The zonal approach without point continuity was previously implemented for specific configurations in the McDonnell Aircraft Company (MCAIR) three-dimensional batch grid generator, ZGRID (Reference 3), and has been used successfully with the MCAIRzonal three-dimensional Euler or Navier-Stokes flow solver, NASTD (Reference 4). Interactive methods provide improved grids by giving the user more direct control of and feedback from the i /

The purpose of this study is to apply shape optimization tools for design of resistance welding electrodes. The numerical simulation of the welding process has been performed by a simplified FEM model implemented in COMSOL. The design process is formulated as an optimization problem where the objective is to prolong the life-time of the electrodes. Welding parameters like current, time and electrode shape parameters are selected to be the design variables while constraints are chosen to ensure a high quality of the welding. Surrogate models based on a Kriging approximation has been used in order to simplify the calculation of shape sensitivities and to generate a generic tool that can be interfaced with other simulation tools. An example numerical study shows the potential of applying optimal design techniques in this area.

Shape optimization in laminated composite plates