Non-Parametric Shape Design of Free-Form Shells Using Fairness Measures and Discrete Differential Geometry

2021 ◽  
Vol 62 (2) ◽  
pp. 93-101
Author(s):  
Makoto Ohsaki ◽  
Kentaro Hayakawa
Keyword(s):  
Differential Geometry ◽  
Optimization Problems ◽  
Geometry Optimization ◽  
Gauss Map ◽  
Discrete Differential Geometry ◽  
Internal Boundary ◽  
Free Form ◽  
Shape Design ◽  
Developable Surface ◽  
Non Parametric

A non-parametric approach is proposed for shape design of free-form shells discretized into triangular mesh. The discretized forms of curvatures are used for computing the fairness measures of the surface. The measures are defined as the area of the offset surface and the generalized form of the Gauss map. Gaussian curvature and mean curvature are computed using the angle defect and the cotangent formula, respectively, defined in the field of discrete differential geometry. Optimization problems are formulated for minimizing various fairness measures for shells with specified boundary conditions. A piecewise developable surface can be obtained without a priori assignment of the internal boundary. Effectiveness of the proposed method for generating various surface shapes is demonstrated in the numerical examples.

scholarly journals Modeling Membrane Dynamics in 3D using Discrete Differential Geometry

2021 ◽  
Vol 120 (3) ◽  
pp. 46a
Author(s):  
Cuncheng Zhu ◽  
Christopher T. Lee ◽  
Ravi Ramamoorthi ◽  
Padmini Rangamani
Keyword(s):  
Differential Geometry ◽  
Membrane Dynamics ◽  
Discrete Differential Geometry

Shape Synthesis and Optimization Using Intrinsic Geometry

1990 ◽  
Author(s):  
Shahriar Tavakkoli ◽  
Sanjay G. Dhande
Keyword(s):  
Optimization Problems ◽  
Piecewise Linear ◽  
Shape Design ◽  
Arc Length ◽  
Intrinsic Geometry ◽  
Shape Model ◽  
Design Variables ◽  
Dimensional Shape ◽  
Variable Geometry Truss ◽  
Arc Lengths

Abstract The present paper outlines a method of shape synthesis using intrinsic geometry to be used for two-dimensional shape optimization problems. It is observed that the shape of a curve can be defined in terms of intrinsic parameters such as the curvature as a function of the arc length. The method of shape synthesis, proposed here, consists of selecting a shape model, defining a set of shape design variables and then evaluating Cartesian coordinates of a curve. A shape model is conceived as a set of continuous piecewise linear segments of the curvature; each segment defined as a function of the arc length. The shape design variables are the values of curvature and/or arc lengths at some of the end-points of the linear segments. The proposed method of shape synthesis and optimization is general in nature. It has been shown how the proposed method can be used to find the optimal shape of a planar Variable Geometry Truss (VGT) manipulator for a pre-specified position and orientation of the end-effector. In conclusion, it can be said that the proposed approach requires fewer design variables as compared to the methods where shape is represented using spline-like functions.

Creation and Manipulation of Complex Displacement Features During the Conceptual Phase of Design

1996 ◽  
Author(s):  
P. A. van Elsas ◽  
J. S. M. Vergeest
Keyword(s):  
Boundary Conditions ◽  
Surface Feature ◽  
Free Form ◽  
Shape Design ◽  
Surface Model ◽  
Test Results ◽  
Base Surface ◽  
Feature Design ◽  
Data Explosion ◽  
Free Form Surface

Abstract Surface feature design is not well supported by contemporary free form surface modelers. For one type of surface feature, the displacement feature, it is shown that intuitive controls can be defined for its design. A method is described that, given a surface model, allows a designer to create and manipulate displacement features. The method uses numerically stable calculations, and feedback can be obtained within tenths of a second, allowing the designer to employ the different controls with unprecedented flexibility. The algorithm does not use refinement techniques, that generally lead to data explosion. The transition geometry, connecting a base surface to a displaced region, is found explicitly. Cross-boundary smoothness is dealt with automatically, leaving the designer to concentrate on the design, instead of having to deal with mathematical boundary conditions. Early test results indicate that interactive support is possible, thus making this a useful tool for conceptual shape design.

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 Hyperbolization of Euclidean Ornaments

10.37236/78 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Martin von Gagern ◽  
Jürgen Richter-Gebert
Keyword(s):  
Differential Geometry ◽  
Pattern Recognition ◽  
Hyperbolic Plane ◽  
Discrete Differential Geometry ◽  
Fundamental Region ◽  
Symmetry Groups ◽  
Symmetry Detection ◽  
Conformal Deformation ◽  
Optimal Resolution ◽  
Wallpaper Groups

In this article we outline a method that automatically transforms an Euclidean ornament into a hyperbolic one. The necessary steps are pattern recognition, symmetry detection, extraction of a Euclidean fundamental region, conformal deformation to a hyperbolic fundamental region and tessellation of the hyperbolic plane with this patch. Each of these steps has its own mathematical subtleties that are discussed in this article. In particular, it is discussed which hyperbolic symmetry groups are suitable generalizations of Euclidean wallpaper groups. Furthermore it is shown how one can take advantage of methods from discrete differential geometry in order to perform the conformal deformation of the fundamental region. Finally it is demonstrated how a reverse pixel lookup strategy can be used to obtain hyperbolic images with optimal resolution.

Non-parametric free-form optimal design of frame structures in natural frequency problem

2016 ◽  
Vol 117 ◽  
pp. 334-345 ◽  
Author(s):  
Masatoshi Shimoda ◽  
Tomohiro Nagano ◽  
Takashi Morimoto ◽  
Yang Liu ◽  
Jin-Xing Shi
Keyword(s):  
Optimal Design ◽  
Natural Frequency ◽  
Free Form ◽  
Frame Structures ◽  
Non Parametric
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