Shape Interpolation Research Articles

Maximizing natural frequencies of inhomogeneous cellular structures by Kriging-assisted multiscale topology optimization

This paper proposes a Kriging-assisted multiscale topology optimization method for maximizing natural frequencies of inhomogeneous cellular structures, where spatially-varying microstructural configurations and their macroscopic distribution are simultaneously optimized. At the beginning, under macroscopic boundary conditions for a cellular structure, the configurations of multiple prototype microstructures are topologically optimized by the parametric level set method (PLSM) combined with the numerical homogenization approach. A kinematical connective constraint is considered to ensure the connectivity between adjacent prototype microstructures. Then, a shape interpolation method is adopted to interpolate shapes of the prototype microstructures, so as to generate a series of sample microstructures. Based on these samples, Kriging metamodels are constructed to predict the effective property of each microstructure within the macrostructure. Finally, the variable thickness sheet (VTS) method is applied to optimize the material distribution pattern at macroscale for maximizing the natural frequency of the cellular structure, where an efficient mode-tracking strategy based on modal assurance criterion (MAC) is employed to track the target mode accurately. Numerical examples are provided to test the performance of the proposed method in natural frequency optimization of cellular structures. The results indicate that the multiscale cellular structures obtained by the proposed method show higher natural frequency compared with the monoscale macrostructural and microstructural designs.

Meshfree simulations of large scale ductile fracture of stiffened ship hull plates during ship stranding

In this work, we have developed a systematic computational methodology to directly simulate the ductile fracture of ship hull structure during ship stranding based on the meshfree Reproducing Kernel Particle Method. The whole computational algorithm has four parts: The first part is establishing interpolation shape functions of RKPM; the second part is the meshfree computational algorithm for simulation of three-dimensional crack propagation; The third part is a computational algorithm for detecting contact and calculating interaction force during ship stranding or grounding, and the last part is the constitutive modeling and update procedure during simulation for ship hull structure material. In this work, we adopt the Gurson-Tvergaard-Needleman model to model large scale ship hull structures. To validate above modeling and computational methodology, a large scale simulation of ship hull structure indentation test is carried out to directly comparing with the experimental test conducted by Alsos and Amdahl (Int J Impact Eng 36(6):799–807, 2009). The comparison between numerical results and experimental test results have been carefully analyzed. Subsequently, the simulations of simplified stranding processes in three different cases are performed. The results from numerical simulations indicate that the proposed computational algorithm is capable of capturing the plastic deformation and damage of ship hull structure subjected to grounding impact.

Topological design of sandwich structures with graded cellular cores by multiscale optimization

Exploring ultralight sandwich structures with superior load-bearing performance is one of the important topics in structural optimization. This paper proposes a novel multiscale topology optimization method to achieve the design of high-performance sandwich structures with graded cellular cores (SSGCCs). In this method, the thicknesses of two solid face-sheets, the graded distribution of cellular sandwich cores at a single layer and their configurations are optimized to well suit for loading conditions, where the single layer is arrayed periodically at its height direction to obtain sandwich layers. Specifically, at macroscale, the variable thickness sheet (VTS) method with the capacity of generating an overall free material distribution pattern, is applied to optimize the thicknesses of two solid face-sheets and achieve the graded distribution of cellular sandwich cores at a single layer. At microscale, a progressive optimization scheme is employed to topologically optimize multiple representative cellular cores (RCCs) at a single layer, so as to achieve their similar topological configurations. With a shape interpolation method, the configurations of graded cellular cores (GCCs) with essential interconnections can be obtained by interpolating the shapes of these RCCs with similar topological features. In order to reduce the computational burden on evaluating effective properties of GCCs by the homogenization method, a Kriging metamodel is constructed based on some key cellular cores as sample points, and adopted to predict the effective properties of all the GCCs. Both 2D and 3D numerical examples are provided to test the validity and advantages of the proposed method for designing SSGCCs.

SDM-NET

We introduce SDM-NET, a deep generative neural network which producesstructured deformable meshes.Specifically, the network is trained to generate a spatial arrangement of closed, deformable mesh parts, which respects the global part structure of a shape collection, e.g., chairs, airplanes, etc. Our key observation is that while the overall structure of a 3D shape can be complex, the shape can usually be decomposed into a set of parts, each homeomorphic to a box, and the finer-scale geometry of the part can be recovered bydeformingthe box. The architecture of SDM-NET is that of atwo-level variational autoencoder(VAE). At the part level, a PartVAE learns a deformable model of part geometries. At the structural level, we train a Structured Parts VAE (SP-VAE), whichjointlylearns the part structure of a shape collection and the part geometries, ensuring the coherence between global shape structure and surface details. Through extensive experiments and comparisons with the state-of-the-art deep generative models of shapes, we demonstrate the superiority of SDM-NET in generating meshes with visual quality, flexible topology, and meaningful structures, benefiting shape interpolation and other subsequent modeling tasks.

Bounded distortion tetrahedral metric interpolation

We present a method for volumetric shape interpolation with unique shape preserving features. The input to our algorithm are two or more 3-manifolds, immersed into R 3 and discretized as tetrahedral meshes with shared connectivity. The output is a continuum of shapes that naturally blends the input shapes, while striving to preserve the geometric character of the input. The basis of our approach relies on the fact that the space of metrics with bounded isometric and angular distortion is convex [Chien et al. 2016b]. We show that for high dimensional manifolds, the bounded distortion metrics form a positive semidefinite cone product space. Our method can be seen as a generalization of the bounded distortion interpolation technique of [Chen et al. 2013] from planar shapes immersed in R 2 to solids in R 3 . The convexity of the space implies that a linear blend of the (squared) edge lengths of the input tetrahedral meshes is a simple yet powerful-and-natural choice. Linearly blending flat metrics results in a new metric which is, in general, not flat, and cannot be immersed into three-dimensional space. Nonetheless, the amount of curvature that is introduced in the process tends to be very low in practical settings. We further design an extremely robust nonconvex optimization procedure that efficiently flattens the metric. The flattening procedure strives to preserve the low distortion exhibited in the blended metric while guaranteeing the validity of the metric, resulting in a locally injective map with bounded distortion. Our method leads to volumetric interpolation with superb quality, demonstrating significant improvement over the state-of-the-art and qualitative properties which were obtained so far only in interpolating manifolds of lower dimensions.

Multiscale topology optimization for minimizing frequency responses of cellular composites with connectable graded microstructures

Abstract This paper develops an efficient multiscale topology optimization method for minimizing the frequency response of cellular composites over a given frequency interval, which consist of spatially-varying connectable graded microstructures. In this method, multiple prototype microstructures are topologically optimized in a progressive manner considering the sensitivity information of two adjacent prototype microstructures, achieving their similar topological configurations. A shape interpolation method is employed to interpolate the shapes of the prototype microstructures to generate a series of graded microstructures from the solid one to void, which are featured with essential interconnections generated by similar shapes of these prototype microstructures. The variable thickness sheet method is adopted at macroscale to generate a free material distribution under a global volume constraint. In order to reduce the computational burden, at microscale, a Kriging metamodel constructed by key graded microstructures as sample points is employed to predict the effective properties of all the microstructures within macrostructure. Furthermore, at macroscale, the quasi-static Ritz vector method is incorporated for efficient frequency response analysis. The method of moving asymptotes is adopted to update the design variables at both scales. In the proposed method, the configurations of spatially-varying connectable graded microstructures at microscale and their distribution at macroscale are simultaneously optimized, which ensures a sufficiently large design space to minimize the frequency response of cellular composites. The topological variations of multiple prototype microstructures further expand this advantage. Numerical examples are provided to test the performance of the proposed multiscale topology optimization method for minimizing the frequency responses of cellular composites.

A shape interpolation procedure: Application to creating explicit grain structure models based on partial data sets

This paper reports a robust shape interpolation procedure for constructing a voxel-based model from a set of parallel 2D sections. The procedure is based on a morphing technique and is applicable to compact shapes of grains that appear in at least two consecutive sections. It relies on a set of images of the same resolution and containing the same sets of colors that correspond to different phases of the given volume. In addition to a simple demo example, we successfully apply the procedure to reconstruct the volumetric grain structure morphology of a thin Al-Mg oligocrystal specimen from two electron-back-scatter-diffraction scans of two parallel faces. For verification purposes the same physical specimen was machined down and two interior sections, parallel to the faces, were scanned. A good matching between reconstructed and actual grain structures at these two sections is observed. As a further illustration, the 3D shapes of several grains are extracted and shown. A demonstration of reconstruction from multiple sections using a dual-phase, martensite-ferrite representative volume element is also given. Finally, a quantitative analysis of false-positive results, as well as a way to decide the applicability of this method for grain reconstruction for a given material are described.

An optimised family of anisotropic microstructures with application to functionally graded materials

We design an anisotropic, multiparametric family of optimised microstructures. The microstructures are parametrised by their Young’s moduli in the three axial directions. The family is smooth, in the sense that the features are similar enough for shape interpolation of the microstructures and interpolation of material properties. We demonstrate how the microstructure family can be used to provide functionally graded materials that match required material property gradients. The microstructure family could also be utilised for multi-scale design of macrosocpic structures with spatially varying microstructure.

An efficient meshless computational technique to simulate nonlinear anomalous reaction–diffusion process in two-dimensional space

In this paper, a numerical simulation of an anomalous reaction–diffusion process in two-dimensional space with a nonlinear source term is presented. An efficient and powerful computational technique based on a combination of a time integration scheme and meshless local Petrov–Galerkin method is implemented for solving the problem. An implicit time stepping scheme with second-order accuracy is used to discretize the problem in the time direction. The unconditional stability of the proposed time discretization scheme is proved in an appropriate Sobolev space. In order to obtain a fully discrete model, the primary spatial domain is represented by a set of regularly distributed nodes and spatial shape functions are constructed on the distributed field nodes using the radial point interpolation method. Then, a local weak form meshless method based on the radial point interpolation shape functions is used to discretize the problem in the spatial direction. The efficiency and accuracy of the proposed computational procedure are demonstrated by numerical examples. Numerical results show that the method approximates solutions of the model in excellent agreement with analytical results. Moreover, it can be observed that the method gives the expected second-order temporal convergence rate independent of the fractional order which confirms the theoretical predictions.

A coupled FE-meshfree method for Helmholtz problems using point interpolation shape functions and edge-based gradient smoothing technique

In this paper, the point interpolation shape functions from a meshfree technique is combined with the generalized gradient smoothing technique (GGST) based on mesh grids to give a coupled FE-meshfree method for analyzing acoustic problems addressed by the Helmholtz equation. It is well-known that the numerical results of Helmholtz problems from several classical numerical approaches, such as the finite element approach, are not always sufficient to provide satisfying accuracy because of the pollution effect for relatively high wave numbers. The pollution effect may generally come from the “overly-stiff” property of the numerical models. Owing to the appropriate softening feature resulting from the edge-based gradient smoothing technique (EGST), the present coupled FE-meshfree method can effectively depress the pollution error effect of the numerical solutions. To avoid the singular issue of the moment matrix for the point interpolation shape functions used, several cell-based T-schemes are utilized for choosing nodes in the interpolation. Numerical results have validated that the present method could not only provide much more accurate solutions compared with the finite element method with the same node distribution, particularly at the relatively high frequency range, but also have higher computational efficiency with certain node selection schemes.

Concurrent topology optimization for cellular structures with nonuniform microstructures based on the kriging metamodel

This paper proposes a novel multiscale concurrent topology optimization for cellular structures with continuously varying microstructures in space to obtain a superior structural performance at an affordable computation cost. At microscale, multiple prototype microstructures are topologically optimized to represent all the microstructures within macrostructure by incorporating a numerical homogenization approach into a parametric level set method (PLSM), whose connectivity is guaranteed by a kinematical connective constraint approach. A shape interpolation technology is developed to map these optimized prototype microstructures and generate a series of nonuniform microstructures, which are considered as sample points and used to construct a kriging metamodel. The built kriging metamodel is then employed to predict the effective properties of all the nonuniform microstructures within macrostructure. At macroscale, the variable thickness sheet (VTS) method is employed to generate an overall free material distribution patterns using the predicted effective properties of all the nonuniform microstructures. With the help of shape interpolation technology, all the nonuniform microstructures within macrostructure are well connected with each other due to the similar topological features at their interfaces. Using the proposed method, the macrostructural topology as well as the locations and configurations of the spatially varying nonuniform microstructures can be simultaneously optimized to ensure a sufficiently large multiscale design space. Numerical examples are provided to demonstrate the validity and advantages of the proposed method.

High-quality compatible triangulations and their application in interactive animation

We propose a new method to compute compatible triangulations of two polygons in order to create smooth geometric transformations between them. Compared to existing methods, our approach creates triangulations of better quality, that is, triangulations with fewer long thin triangles and Steiner points. This results in visually appealing morphings when transforming the shape from one into another. Our method consists of three stages. First, we use a common valid vertex pair to uniquely decompose the source and target polygons into pairs of sub-polygons, in which each concave sub-polygon is triangulated. Second, within each sub-polygon pair, we map the triangulation of a concave sub-polygon onto the corresponding sub-polygon using a linear transformation, thereby generating compatible meshes between the source and the target. Third, we refine the compatible meshes, which creates better quality planar shape morphing with detailed textures. In order to evaluate the quality of the resulting mesh, we present a new metric that assesses the deformation of each triangle during the shape morphing process. Finally, we present an efficient scheme to handle compatible triangulations for a shape with self-occlusion, resulting in an interactive shape morphing system. Experimental results show that our method can create compatible meshes of higher quality as compared to existing methods with fewer long thin triangles and smaller triangle deformation values during shape morphing. These advantages enable us to create more consistent rotations for rigid shape interpolation algorithms and facilitate a smoother morphing process. The proposed algorithm is both robust and computationally efficient. It can be applied to produce convincing transformations such as interactive 2D animation and texture mapping. The proposed interactive shape morphing system enables normal users to generate morphing video easily without any professional knowledge.

The solution of nonlinear Green–Naghdi equation arising in water sciences via a meshless method which combines moving kriging interpolation shape functions with the weighted essentially non–oscillatory method

In this investigation a new meshless numerical technique is proposed for solving Green–Naghdi equation by combining the moving Kriging interpolation shape functions with the weighted essentially non-oscillatory (WENO) method. The present approach has been taken from [12, 30]. The convergence order of WENO technique can be studied by the number of interpolation nodes because this method is described by interpolation concept. The proposed method is based on the non-polynomial WENO procedure in order to increase the convergence order and local accuracy. Four examples have been solved that they show the efficiency and accuracy of the proposed method.

Deformation-induced surface roughening of an Al-Mg alloy

We analyze the deformation-induced surface roughening of an Al-Mg alloy. One aspect of our work is an effort to model the exact roughening behavior of a mesoscale-size specimen that contains a small number of grains. Such a specimen is obtained using prestrain, recrystallization annealing and subsequent machining. It contains one layer of grains through the thickness and is tested under uniaxial tension. A laser confocal microscope is then used in order to measure the surface profile of a deformed specimen. We apply crystal plasticity finite element method to a voxel model of a given specimen. The exact arrangement of grains in the voxel model is reconstructed from a pair of electron backscatter diffraction (EBSD) scans using a custom-developed shape interpolation procedure. The material properties of the Al-Mg alloy are found using an efficient black-box optimizer. The exact shape of the deformed specimen wasn’t fully captured, however results on average surface roughness show good matching between model and experiment.

A multiobjective sensor placement optimization for SHM systems considering Fisher information matrix and mode shape interpolation

Sensor placement optimization plays a key role in structural health monitoring (SHM) of large mechanical structures. Given the existence of an effective damage identification procedure, the problem arises as to how the acquisition points should be placed for optimal efficiency of the detection system. The global multiobjective optimization of sensor locations for structural health monitoring systems is studied in this paper. First, a laminated composite plate is modelled using Finite Element Method (FEM) and put into modal analysis. Then, multiobjective genetic algorithms (GAs) are adopted to search for the optimal locations of sensors. Numerical issues arising in the selection of the optimal sensor configuration in structural dynamics are addressed. A method of multiobjective sensor locations optimization using the collected information by Fisher Information Matrix (FIM) and mode shape interpolation is presented in this paper. The sensor locations are prioritized according to their ability to localize structural damage based on the eigenvector sensitivity method. The proposed method presented in this paper allows to distribute the points of acquisition on a structure in the best possible way so as to obtain both data of greater modal information and data for better modal reconstruction from a minimum point interpolation. Numerical example and test results show that the proposed method is effective to distribute a reduced number of sensors on a structure and at the same time guarantee the quality of information obtained. The results still indicate that the modal configuration obtained by multiobjective optimization does not become trivial when a set of modes is used in the construction of the objective function. This strategy is an advantage in experimental modal analysis tests, since it is only necessary to acquire signals in a limited number of points, saving time and operational costs.

Pricing real estate index options by compactly supported radial-polynomial basis point interpolation

This paper presents a novel method to price the real estate index options, which are modeled based on the framework proposed by Fabozzi et al. (2012). The CS–PBF method combines compactly supported radial basis functions (CSRBF) and polynomial basis functions (PBF) to yield the interpolation functions, which can guarantee interpolation shape functions with Kronecker property and overcome possible singularity associated with the PBF method. Compared with the CSRBF method and the finite difference (FD) method, the CS–PBF method is more accurate and efficient for the real estate index option. Meanwhile, a local mesh refinement technique is employed for dealing with the non-smooth options’ payoffs, which is very effective and stable to improve the computational accuracy for the CS–PBF method. Finally, the CS–PBF method is extended to price American option of the real estate index.

Minimization of Poisson’s ratio in anti-tetra-chiral two-phase structure

One of the most important goal of modern material science is designing structures which exhibit appropriate properties. These properties can be obtained by optimization methods which often use numerical calculations e.g. finite element method (FEM). This paper shows the results of topological optimization which is used to obtain the greatest possible negative Poisson’s ratio of the two-phase composite. The shape is anti-tetra-chiral two-dimensional unit cell of the whole lattice structure which has negative Poisson’s ratio when it is built of one solid material. Two phase used in optimization are two solid materials with positive Poisson’s ratio and Young’s modulus. Distribution of reinforcement hard material inside soft matrix material in anti-tetra-chiral domain influenced mechanical properties of structure. The calculations shows that the resultant structure has negative Poisson’s ratio even eight times smaller than homogenous anti-tetra chiral structure made of classic one material. In the analysis FEM is connected with algorithm Method of Moving Asymptote (MMA). The results of materials’ properties parameters are described and calculated by means of shape interpolation scheme – Solid Isotropic Material with Penalization (SIMP) method.

Deblurring and Denoising of Maps between Shapes

AbstractShape correspondence is an important and challenging problem in geometry processing. Generalized map representations, such as functional maps, have been recently suggested as an approach for handling difficult mapping problems, such as partial matching and matching shapes with high genus, within a generic framework. While this idea was shown to be useful in various scenarios, such maps only provide low frequency information on the correspondence. In many applications, such as texture transfer and shape interpolation, a high quality pointwise map that can transport high frequency data between the shapes is required. We name this problem map deblurring and propose a robust method, based on a smoothness assumption, for its solution. Our approach is suitable for non‐isometric shapes, is robust to mesh tessellation and accurately recovers vertex‐to‐point, or precise, maps. Using the same framework we can also handle map denoising, namely improvement of given pointwise maps from various sources. We demonstrate that our approach outperforms the state‐of‐the‐art for both deblurring and denoising of maps on benchmarks of non‐isometric shapes, and show an application to high quality intrinsic symmetry computation.

Orbit Segmentation by Surface Reconstruction With Automatic Sliced Vertex Screening.

The purpose of this paper is to develop a computational approach to the segmentation of human orbits. The first step is to perform Hounsfield units thresholding to segment the bony structure around the orbit. Then, a three-dimensional mesh model is generated. Poisson surface reconstruction is applied to a set of automatically screened vertices, which are facing the inner orbital walls. These procedures effectively close orbital fissures; various nerves foramina; and interpolate the broken surfaces due to thin bone structures around the orbit. We also developed validation models with five dried skulls, where the orbits were filled with dental impression. Validations on the proposed algorithm were performed with the corresponding CT images and verified by experienced radiographer. The mean volume differences are less than 0.3%. Surface differences are within 0.3 mm of root mean square. Both differences are not clinically significant. Traditional approaches are slice-by-slice manual editing or shape interpolation with selected slices interactively. It is not only time consuming, but also inefficient, exhibits interoperator variability, and repeatability problems. In the proposed method, most of the manual processes are eliminated with adjustable vertex screening parameters. It makes the proposed method repeatable.

Physically Realizable Three-Dimensional Bone Prosthesis Design With Interpolated Microstructures.

We present a new approach to designing three-dimensional, physically realizable porous femoral implants with spatially varying microstructures and effective material properties. We optimize over a simplified design domain to reduce shear stress at the bone-prosthetic interface with a constraint on the bone resorption measured using strain energy. This combination of objective and constraint aims to reduce implant failure and allows a detailed study of the implant designs obtained with a range of microstructure sets and parameters. The microstructure sets are either specified directly or constructed using shape interpolation between a finite number of microstructures optimized for multifunctional characteristics. We demonstrate that designs using varying microstructures outperform designs with a homogeneous microstructure for this femoral implant problem. Further, the choice of microstructure set has an impact on the objective values achieved and on the optimized implant designs. A proof-of-concept metal prototype fabricated via selective laser melting (SLM) demonstrates the manufacturability of designs obtained with our approach.