On the Efficiency of a Point Cloud Autoencoder as a Geometric Representation for Shape Optimization

We present an approach for shape optimization of large-scale aerodynamic problems, combining free-form deformation and POD based reduced-order modeling. An extension of the classical Free-Form Deformation techniques is derived in order to handle efficiently constraints regarding fixed and deformable portions of the geometry and to impose the smoothness at the interface between the two different regions. The second aspect concerns the development of a hybrid model, combining a POD-based reduced order model and a industrial CFD solver using a domain-decomposition approach. A method on how determining automatically a suitable domain decomposition is discussed. The effectiveness and drawback of the above techniques are highlighted on a large-scale aerodynamic shape optimization and control problem, i.e. the mainsail thrust optimization of a sailing boat.

Aerodynamic shape optimization of a single turbine stage based on parameterized Free-Form Deformation with mapping design parameters

Profile extrusion is a continuous production process for manufacturing plastic profiles from molten polymer. Especially interesting is the design of the die, through which the melt is pressed to impart the desired shape. However, due to an inhomogeneous velocity distribution at the die exit or residual stresses inside the extrudate, the final shape of the manufactured part often deviates from the desired one [1]. To avoid these deviations, we want to optimize the shape of the flow channel inside the die computationally. This has already been investigated in the literature using conventional optimization approaches [2,3].In this work, we investigate the feasibility of Reinforcement Learning (RL) as a learning‐based approach for shape optimization. RL is based on trial‐and‐error interactions of an agent with an environment. For each action, the agent is rewarded and informed about the subsequent state of the environment, which for this application stems from a high‐fidelity Finite Element Method (FEM) simulation.We will introduce a 2D test case as a proof‐of‐concept, in which an RL agent learns to change the geometry of a flow channel by modifying its representation as computational mesh through a spline‐based deformation method known as Free Form Deformation (FFD) [4]. We will show that an agent can be trained to optimize the geometry for different values of the quantity of interest and that the learning behavior is reproducible, which renders the RL‐based approach promising for further research.

In this paper we consider a shape optimization problem in which the data in the cost functional and in the state equation may change sign, and so no monotonicity assumption is satisfied. Nevertheless, we are able to prove that an optimal domain exists. We also deduce some necessary conditions of optimality for the optimal domain. The results are applied to show the existence of an optimal domain in the case where the cost functional is completely identified, while the right-hand side in the state equation is only known up to a probability $P$ in the space $L^2(D)$.

An improved marine predators algorithm for shape optimization of developable Ball surfaces

Two approaches are proposed for solving inverse problems in shape optimization. We are looking for the unknown position of a small hole in a domain Ω. First, the asymptotic analysis of the underlying p.d.e. defined in a perturbed domain is performed and the so-called topological derivative is defined. Then, in the first approach, the self-adjoint extensions of elliptic operators are used to model the solution of a partial differential equation defined in the singularly perturbed domain. A least-square functional is then minimized to identify the hole. In the second approach, neural networks are used to determine the inverse of the mapping which associates a set of shape functionals to the position of the unknown hole. In both approaches the topological derivatives are used to approximate the shape functionals.

The shape optimization problems naturally appear in engineering and biology. They aim to answer questions as:-What a perfect wing may look like?-How to minimize the resistance of a moving object in a gas or a fluid?-How to build a rod of maximal rigidity?-What is the behaviour of a system of cells?The shape optimization appears also in physics, mainly in electrodynamics and in the systems presenting both classical and quantum mechanics behaviour. For explicit examples and furtheraccount on the applications of the shape optimization we refer to the books [20] and [69]. Here we deal with the theoretical mathematical aspects of the shape optimization, concerning existence of optimal sets and their regularity. In all the practical situations above, the shape of the object in study is determined by a functional depending on the solution of a given partial differential equation. We will sometimes refer to this function as a state function.The simplest state functions are provided by solutions of the equations−∆w = 1 and −∆u = λu,which usually represent the torsion rigidity and the oscillation modes of a given object. Thus our study will be concentrated mainly on the situations, in which these state functions appear,i.e. when the optimality is intended with respect to energy and spectral functionals. [20] D. Bucur, G. Buttazzo: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations 65, Birkhauser Verlag, Basel (2005).[69] A. Henrot, M. Pierre: Variation et optimisation de formes: une analyse geometrique. Springer-Berlag, Berlin, 2005.

Free-Form Deformation (FFD) is an efficient technique for editing the shapes of 3D models, and widely used in the Battlefield Environment Visualization (BEV), computer animation, computer graphics entertainment etc. The basic idea of some famous previous FFD approaches deforms a target 3D model by adjusting some control points of a 3D lattice surrounding the model. It is a tedious work, especially when the lattice contains too many control points. Moreover, how to place the control points of the lattice to make them cover the region of the target model is also a problem, and it is also difficult to keep the geometric measurement for the deformed model. Therefore, in this paper, a novel FFD method without any lattice like structure is proposed by borrowing the idea of non-distortion texture-mapping for freeform surfaces. By using this method, the 3D non-regular military symbol of the deformed model due to a given parametric terrain surface can be predicted easily, so that the user can get his or her desirable results more intuitively and has no necessary to place the control points of a proper lattice. Moreover, since this method is simple, efficient, and has a real-time response, it is more suitable for doing the animation with 3D non-regular military symbol models in the battlefield terrain surface.

An enhanced approach for shape optimization using an adaptive algorithm

Aerodynamic shape optimization plays a fundamental role in aircraft design. However, useful parameterizations of shapes for engineering models often result in high-dimensional design spaces which can create challenges for both local and global optimizers. In this paper, we employ an active subspace method (ASM) to discover and exploit low-dimensional, monotonic trends in the quantity of interest as a function of the design variables. The trend enables us to eciently and eectively nd an optimal design in appropriate areas of the design space. We apply this approach to the ONERA-M6 transonic wing, parameterized with 50 Free-Form Deformation (FFD) design variables. Given an initial set of 300 designs, the ASM discovered a low-dimensional linear subspace of the input space that explained the majority of the variability in the drag and lift coecients. This revealed a global trend that we exploited to nd an optimal design with reduced computational cost.

Multi-objective optimization of a high-speed train head based on the FFD method

Parametric Control of the Hydraulic Machinery Impeller Based on Free-Form Deformation

This study was undertaken to register 3D parametric breast images derived from Gd-DTPA MR and F-18-FDG PET/CT dynamic image series. Nonlinear curve fitting (Levenburg-Marquardt algorithm) based on realistic two-compartment models was performed voxel-by-voxel separately for MR (Brix) and PET (Patlak). PET dynamic series consists of 50 frames of 1-minute duration. Each consecutive PET image was nonrigidly registered to the first frame using a finite element method and fiducial skin markers. The 12 post-contrast MR images were nonrigidly registered to the precontrast frame using a free-form deformation (FFD) method. Parametric MR images were registered to parametric PET images via CT using FFD because the first PET time frame was acquired immediately after the CT image on a PET/CT scanner and is considered registered to the CT image. We conclude that nonrigid registration of PET and MR parametric images using CT data acquired during PET/CT scan and the FFD method resulted in their improved spatial coregistration. The success of this procedure was limited due to relatively large target registration error, TRE = 15.1±7.7 mm, as compared to spatial resolution of PET (6-7 mm), and swirling image artifacts created in MR parametric images by the FFD. Further refinement of nonrigid registration of PET and MR parametric images is necessary to enhance visualization and integration of complex diagnostic information provided by both modalities that will lead to improved diagnostic performance.

Multi-objective free-form optimization for shape and thickness of shell structures with composite materials

Geometric deep learning has been revolutionizing the molecular modeling field. Despite the state-of-the-art neural network models are approaching ab initio accuracy for molecular property prediction, their applications, such as drug discovery and molecular dynamics (MD) simulation, have been hindered by insufficient utilization of geometric information and high computational costs. Here we propose an equivariant geometry-enhanced graph neural network called ViSNet, which elegantly extracts geometric features and efficiently models molecular structures with low computational costs. Our proposed ViSNet outperforms state-of-the-art approaches on multiple MD benchmarks, including MD17, revised MD17 and MD22, and achieves excellent chemical property prediction on QM9 and Molecule3D datasets. Furthermore, through a series of simulations and case studies, ViSNet can efficiently explore the conformational space and provide reasonable interpretability to map geometric representations to molecular structures.