Seamless integration of analysis and design – Automatic CAD reconstruction of post-analysis geometries

Sebastian Hube; Roxana Pohlmann; Stefanie Elgeti

doi:10.3934/acse.2023007

Article Contents

2023, Volume 1, Issue 2: 162-179. Doi: 10.3934/acse.2023007

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Seamless integration of analysis and design – Automatic CAD reconstruction of post-analysis geometries

1.
Chair for Computational Analysis of Technical Systems, RWTH Aachen University, Germany
2.
Institute of Lightweight Design and Structural Biomechanics, TU Wien, Austria

^*Corresponding author: Roxana Pohlmann
^*Corresponding author: Roxana Pohlmann

Received: May 2023

Revised: June 2023

Early access: June 2023

Published: June 2023

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Abstract

A key step during industrial design is the passing of design information from computer aided design (CAD) to analysis tools (CAE) and vice versa. Here, one is faced with a severe incompatibility in geometry representation: While CAD is usually based on surface representations, analysis mostly relies on volumetric representations. The forward pass, i.e., converting CAD data to computational meshes, is well understood and established. However, the same does not hold for the inverse direction, i.e., CAD reconstruction of deformed geometries resulting from analysis. This is particularly important for industrial workflows in which the shape optimization of an initial product is outsourced. Such shape optimization is the focus of this work. The few reconstruction methods reported mainly rely on spline fitting, in particular on creating new splines similar to shape reconstruction from 3D imaging. In contrast, this paper studies a novel approach that reuses the CAD data given in the initial design. We show that this concept enables one to shape reconstruct mediocre deformations while preserving the initial notion of features defined during design. Furthermore, reusing the initial CAD representation reduces the shape reconstruction problem to a shape modification problem. We study this unique feature and show that it enables the reconstruction of CAD data from computational meshes by composing each spline in the initial CAD data with a single, global deformation spline. While post-processing is needed for use in current CAD software, most notably, this novel approach enables reconstructing complete CAD models even from defeatured computational meshes.

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Mathematics Subject Classification: Primary: 62P30, 68U07; Secondary: 65D07, 65D17.

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Figure 1. Shape optimization of the initial design (left) modifies the boundaries of existing geometric features (right). Modified from [21]

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Figure 2. Topology optimization of the design space (left) determines the material distribution within the design space (right). Modified from [21]

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Figure 3. Shape optimization in an industrial design workflow from top to bottom: For efficient computations, the CAD model is defeatured, and a computational mesh is obtained by tesselating this defeatured model. Shape-optimizing CAE yields a deformed computational mesh and this deformation needs to be imposed onto the CAD model

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Figure 4. Topological connection of three single surfaces to a closed cylinder in CAD data. (A) shows the geometric representation of the single surfaces, (B) the intersected visualized parts, and (C) their topological connection as a surface representation. Modified from [19]

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Figure 5. We embed an initial point cloud (A) in a spline's parameter space (B). By adapting the control points, the trivariate mapping approximates the deformation between initial and shape-optimized point cloud (C)

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Figure 6. Composition of a sphere and a deformation spline. The initial sphere template is bi-quadratic. The resulting composed sphere has degrees $ o = 12 $ and $ p = 12 $ along its parametric dimensions

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Figure 7. The physical space of a disk (pink) is contained within the parametric space of another surface spline (green)

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Figure 8. The mapping $ \boldsymbol{\mathcal{{T}}} $ is defined on the defeatured computational mesh without the stiffeners (A), and applied to the CAD model (B) resulting in updated CAD data. The approach yields watertight reconstructions even if defeatured models are used in CAE (compare missing stringers)

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Figure 9. Initial geometry of plate with hole shown with different color for each entity in the geometry data. Results will show the identical coloring for identical entities. The dimensions are $ 200\ {\rm{mm}} \times 100\ {\rm{mm}} \times 1.5\ {\rm{mm}} $ and the hole diameter is $ 50\ {\rm{mm}} $

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Figure 10. Functional composition applied to a simple geometry

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Figure 11. The proposed method deforms an initial CAD model (A) to the shape of a deformed computational grid (B)

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Figure 12. Resulting reconstructed shape with degree-reduction

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Table 1. The reconstruction error for the plate with a hole (in mm) of (A): the composed model, and (B): the composed and degree-reduced model (cf. Sec 4.3)

	Min. error	Max. error	Mean error	RMS
A	$ 7.5102 \times 10^{-3} $	$ 1.0764 $	$ 2.9600 \times 10^{-1} $	$ 3.5253 \times 10^{-1} $
B	$ 7.5103 \times 10^{-3} $	$ 1.0764 $	$ 2.9600 \times 10^{-1} $	$ 3.5254 \times 10^{-1} $

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Table 2. Computational time (in s) for the plate with a hole. Each major processing step (functional composition, trivariate fit and low-degree approximation) is individually timed for each single spline $ \boldsymbol{\mathcal{{S}}}^i \in \Sigma $. A comparison is given by individually fitting each spline $ \boldsymbol{\mathcal{{S}}}^i $ instead of using a global deformation spline (Individual splinefit)

Process step	Cumulative	Minimal	Maximal
Splinefit	$ 20.000 $	–	–
FC	$ 0.564 $	6.664 × 10⁻³	6.438 × 10⁻²
Low-degree approx.	3.139	6.610 × 10⁻²	0.657
Overall	$ 23.703 $	–	–
Individual splinefit (reference)	4.304 × 10⁴	$ {20.010} $	1.746 × 10⁴

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Table 3. The reconstruction error (in mm) for the pressure sensor of: (A) composed model, (B) composed and degree-reduced model (cf. Sec 4.3)

	Min. error	Max. error	Mean error	RMS
A	$ 3.3194 \times 10^{-5} $	$ 4.5816 \times 10^{-2} $	$ 3.5244 \times 10^{-3} $	$ 5.2402 \times 10^{-3} $
B	$ 3.3213 \times 10^{-5} $	$ 4.5816 \times 10^{-2} $	$ 3.5373 \times 10^{-3} $	$ 5.2446 \times 10^{-3} $

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Table 4. Computational time (in s) for the pressure sensor with our method at each individual step and overall. Similar to Tab. 2, timings are compared to an individual splinefit of each curve or surface spline in the geometry (Individual splinefit)

Process step	Cumulative	Minimal	Maximal
Global splinefit	5.785 × 10³	–	–
FC	16.120	1.094 × 10⁻³	7.839 × 10⁻²
Low-degree approx.	4.343 × 10⁴	6.340	6.341 × 10²
Overall	4.923 × 10⁴	–	–
Individual splinefit (reference)	4.192 × 10⁵	20.010	1.166 × 10⁴

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