Identification of spring parameters for deformable object simulation

Mass spring models are frequently used to simulate deformable objects because of their conceptual simplicity and computational speed. Unfortunately, the model parameters are not related to elastic material constitutive laws in an obvious way. Several methods to set optimal parameters have been proposed but, so far, only with limited success. We analyze the parameter identification problem and show the difficulties, which have prevented previous work from reaching wide usage. Our main contribution is a new method to derive analytical expressions for the spring parameters from an isotropic linear elastic reference model. The method is described and expressions for several mesh topologies are derived. These include triangle, rectangle, and tetrahedron meshes. The formulas are validated by comparing the static deformation of the MSM with reference deformations simulated with the finite element method.     

Remarks on rate constitutive equations for finite deformation problems: computational implications

It is explicitly shown that if the (spatial) elasticity tensor of an elastic material is taken as isotropic for all possible configurations, then its coefficients cannot be constants; they must depend nontrivially on the Jacobian determinant of the deformation gradient. Moreover, the assumption typically made for computational purposes that its coefficients remain constant for all possible configurations is incompatible with elasticity. It is further shown that an assumption widely used in the computational literature in the context of finite deformation plasticity, namely, relating an objective stress rate to the rate of deformation tensor through a fourth-rank constant isotropic tensor, is also incompatible with elasticity, thus furnishing an example of an hypoelastic material which is not elastic.     

Shape of Elastic Strings in Euclidean Space

We construct a 1-parameter family of geodesic shape metrics on a space of closed parametric curves in Euclidean space of any dimension. The curves are modeled on homogeneous elastic strings whose elasticity properties are described in terms of their tension and rigidity coefficients. As we change the elasticity properties, we obtain the various elastic models. The metrics are invariant under reparametrizations of the curves and induce metrics on shape space. Analysis of the geometry of the space of elastic strings and path spaces of elastic curves enables us to develop a computational model and algorithms for the estimation of geodesics and geodesic distances based on energy minimization. We also investigate a curve registration procedure that is employed in the estimation of shape distances and can be used as a general method for matching the geometric features of a family of curves. Several examples of geodesics are given and experiments are carried out to demonstrate the discriminative quality of the elastic metrics.     

Sample re-weighting hyper box classifier for multi-class data classification

In this work, we propose two novel classifiers for multi-class classification problems using mathematical programming optimisation techniques. A hyper box-based classifier (Xu & Papageorgiou, 2009) that iteratively constructs hyper boxes to enclose samples of different classes has been adopted. We firstly propose a new solution procedure that updates the sample weights during each iteration, which tweaks the model to favour those difficult samples in the next iteration and therefore achieves a better final solution. Through a number of real world data classification problems, we demonstrate that the proposed refined classifier results in consistently good classification performance, outperforming the original hyper box classifier and a number of other state-of-the-art classifiers.Furthermore, we introduce a simple data space partition method to reduce the computational cost of the proposed sample re-weighting hyper box classifier. The partition method partitions the original dataset into two disjoint regions, followed by training sample re-weighting hyper box classifier for each region respectively. Through some real world datasets, we demonstrate the data space partition method considerably reduces the computational cost while maintaining the level of prediction accuracies.     

An Element Free Galerkin Method Based on the Modified Moving Least Squares Approximation

This paper demonstrates that the recently developed modified moving least squares (MMLS) approximation possess the necessary properties which allow its use as an element free Galerkin (EFG) approximation method. Specifically, the consistency and invariance properties for the MMLS are proven. We demonstrate that MMLS shape functions form a partition of unity and the MMLS approximation satisfies the patch test. The invariance properties are important for the accurate computation of the shape functions by using translation and scaling to a canonical domain. We compare the performance of the EFG method based on MMLS, which uses quadratic base functions, to the performance of the EFG method which uses classical MLS with linear base functions, using both 2D and 3D examples. In 2D we solve an elasticity problem which has an analytical solution (bending of a Timoshenko beam) while in 3D we solve an elasticity problem which has an exact finite element solution (unconstrained compression of a cube). We also solve a complex problem involving complicated geometry, non-linear material, large deformations and contacts. The simulation results demonstrate the superior performance of the MMLS over classical MLS in terms of solution accuracy, while shape functions can be computed using the same nodal distribution and support domain size for both methods.     

Axisymmetric bending of circular and annular sandwich plates with nonlinear elastic core material

This paper compares the analytical model of the axisymmetric bending of a circular sandwich plate with the finite element method (FEM) based numerical model. The differential equations of the bending of circular symmetrical sandwich plates with isotropic face sheets and a nonlinear elastic core material are obtained. The perturbation method of a small parameter is used to represent the nonlinear differential equations as a sequence of linear equations specifying each other. The linear differential equations are solved by reducing them to the Bessel equation. The results of the calculations with the use of the analytical and FEM models are compared with the results obtained by other authors by the example of the following problems: (1) axisymmetric transverse bending of a circular sandwich plate; (2) axisymmetric transverse bending of an annular sandwich plate. The effect of the nonlinear elasticity of the core material on the strained state of the sandwich plate is described.     

Physical modeling with orthotropic material based on harmonic fields

Although it is well known that human bone tissues have obvious orthotropic material properties, most works in the physical modeling field adopted oversimplified isotropic or approximated transversely isotropic elasticity due to the simplicity. This paper presents a convenient methodology based on harmonic fields, to construct volumetric finite element mesh integrated with complete orthotropic material. The basic idea is taking advantage of the fact that the longitudinal axis direction indicated by the shape configuration of most bone tissues is compatible with the trajectory of the maximum material stiffness. First, surface harmonic fields of the longitudinal axis direction for individual bone models were generated, whose scalar distribution pattern tends to conform very well to the object shape. The scalar iso-contours were extracted and sampled adaptively to construct volumetric meshes of high quality. Following, the surface harmonic fields were expanded over the whole volumetric domain to create longitudinal and radial volumetric harmonic fields, from which the gradient vector fields were calculated and employed as the orthotropic principal axes vector fields. Contrastive finite element analyses demonstrated that elastic orthotropy has significant effect on simulating stresses and strains, including the value as well as distribution pattern, which underlines the relevance of our orthotropic modeling scheme.     

Improved 2D mass-spring-damper model with unstructured triangular meshes

In this paper, we investigate both the visual realism and the physical accuracy of the 2D mass-spring-damper (MSD) model with general unstructured triangular meshes for the simulation of rigid cloth. For visual realism, the model should, at a minimum, bend smoothly under pure bending load conditions. For physical accuracy, it should bend approximately the same amount and shape as dictated by continuum mechanics. By matching the 2D MSD model with an elastic plate, we obtain a series of constraints on the parameters of the model. We find that for a 2D unstructured MSD model, it is necessary to apply preloads on the springs for accurate modeling of bending resistance. By simultaneously applying the constraints for both visual realism and physical accuracy, we can optimize the parameters of the model to enhance its fidelity. The simulation shows that the deformation of the optimized MSD model with preload is very close to the result obtained by the finite element method (FEM) under either point load condition or pressure load condition. With a much smaller computational burden compared with FEM, the optimized MSD model is especially suitable for real time haptic applications.     

Elastic Correspondence between Triangle Meshes

We propose a novel approach for shape matching between triangular meshes that, in contrast to existing methods, can match crease features. Our approach is based on a hybrid optimization scheme, that solves simultaneously for an elastic deformation of the source and its projection on the target. The elastic energy we minimize is invariant to rigid body motions, and its non‐linear membrane energy component favors locally injective maps. Symmetrizing this model enables feature aligned correspondences even for non‐isometric meshes. We demonstrate the advantage of our approach over state of the art methods on isometric and non‐isometric datasets, where we improve the geodesic distance from the ground truth, the conformal and area distortions, and the mismatch of the mean curvature functions. Finally, we show that our computed maps are applicable for surface interpolation, consistent cross‐field computation, and consistent quadrangular remeshing of a set of shapes.     

Generic remeshing of 3D triangular meshes with metric-dependent discrete voronoi diagrams

In this paper, we propose a generic framework for 3D surface remeshing. Based on a metric-driven Discrete Voronoi Diagram construction, our output is an optimized 3D triangular mesh with a user defined vertex budget. Our approach can deal with a wide range of applications, from high quality mesh generation to shape approximation. By using appropriate metric constraints the method generates isotropic or anisotropic elements. Based on point-sampling, our algorithm combines the robustness and theoretical strength of Delaunay criteria with the efficiency of entirely discrete geometry processing . Besides the general described framework, we show experimental results using isotropic, quadric-enhanced isotropic and anisotropic metrics which prove the efficiency of our method on large meshes, for a low computational cost.     

A numerical investigation on the interplay amongst geometry, meshes, and linear algebra in the finite element solution of elliptic PDEs

In this paper, we study the effect of the choice of mesh quality metric, preconditioner, and sparse linear solver on the numerical solution of elliptic partial differential equations (PDEs). We smooth meshes on several geometric domains using various quality metrics and solve the associated elliptic PDEs using the finite element method. The resulting linear systems are solved using various combinations of preconditioners and sparse linear solvers. We use the inverse mean ratio and radius ratio metrics in addition to conditioning-based scale-invariant and interpolation-based size-and-shape metrics. We employ the Jacobi, SSOR, incomplete LU, and algebraic multigrid preconditioners and the conjugate gradient, minimum residual, generalized minimum residual, and bi-conjugate gradient stabilized solvers. We focus on determining the most efficient quality metric, preconditioner, and linear solver combination for the numerical solution of various elliptic PDEs with isotropic coefficients. We also investigate the effect of vertex perturbation and the effect of increasing the problem size on the number of iterations required to converge and on the solver time. In this paper, we consider Poisson’s equation, general second-order elliptic PDEs, and linear elasticity problems.     

An Elasticity-Based Covariance Analysis of Shapes

We introduce the covariance of a number of given shapes if they are interpreted as boundary contours of elastic objects. Based on the notion of nonlinear elastic deformations from one shape to another, a suitable linearization of geometric shape variations is introduced. Once such a linearization is available, a principal component analysis can be investigated. This requires the definition of a covariance metric—an inner product on linearized shape variations. The resulting covariance operator robustly captures strongly nonlinear geometric variations in a physically meaningful way and allows to extract the dominant modes of shape variation. The underlying elasticity concept represents an alternative to Riemannian shape statistics. In this paper we compare a standard L ²-type covariance metric with a metric based on the Hessian of the nonlinear elastic energy. Furthermore, we explore the dependence of the principal component analysis on the type of the underlying nonlinear elasticity. For the built-in pairwise elastic registration, a relaxed model formulation is employed which allows for a non-exact matching. Shape contours are approximated by single well phase fields, which enables an extension of the method to a covariance analysis of image morphologies. The model is implemented with multilinear finite elements embedded in a multi-scale approach. The characteristics of the approach are demonstrated on a number of illustrative and real world examples in 2D and 3D.     

Hybrid-displacement (Trefftz) formulation for softening materials

This paper reports on the nonlinear static analysis of 2D concrete structures using a non-conventional finite element formulation. The nonlinear behavior of the material is modelled with a continuum non-local and isotropic damage model. While the material’s behavior is linear elastic, a pure hybrid-displacement Trefftz formulation is adopted. From the point where the concrete assumes a nonlinear behavior, this approach degenerates into a hybrid-displacement formulation. The computational performance is tested by means of two numerical examples which show that the proposed model predicts correctly the global behavior of the structures.     

Real-time elastic deformations of soft tissues for surgerysimulation

We describe a novel method for surgery simulation including a volumetric model built from medical images and an elastic modeling of the deformations. The physical model is based on elasticity theory which suitably links the shape of deformable bodies and the forces associated with the deformation. A real time computation of the deformation is possible thanks to a preprocessing of elementary deformations derived from a finite element method. This method has been implemented in a system including a force feedback device and a collision detection algorithm. The simulator works in real time with a high resolution liver model     

Thermoelastic deformations in graphene and analogous two-dimensional materials

Using the Duhamel–Neumann equations, we consider the stationary heat-loading problem of a bulk specimen of a two-dimensional material (like grapheme) as an approximation of small elastic deformations. We present a numerical method for solving the heat-loading problem of a specimen of a complex shape with the use of a Friedrichs-monotonic finite-difference scheme on chaotic grids in a multiply connected integration domain. Then we demonstrate the results of the computational experiments.     

Adapting Feature Curve Networks to a Prescribed Scale

Feature curves on surface meshes are usually defined solely based on local shape properties such as dihedral angles and principal curvatures. From the application perspective, however, the meaningfulness of a network of feature curves also depends on a global scale parameter that takes the distance between feature curves into account, i.e., on a coarse scale, nearby feature curves should be merged or suppressed if the surface region between them is not representable at the given scale/resolution. In this paper, we propose a computational approach to the intuitive notion of scale conforming feature curve networks where the density of feature curves on the surface adapts to a global scale parameter. We present a constrained global optimization algorithm that computes scale conforming feature curve networks by eliminating curve segments that represent surface features, which are not compatible to the prescribed scale. To demonstrate the usefulness of our approach we apply isotropic and anisotropic remeshing schemes that take our feature curve networks as input. For a number of example meshes, we thus generate high quality shape approximations at various levels of detail.     

A local tangential lifting differential method for triangular meshes

In this note we present a local tangential lifting (LTL) algorithm to compute differential quantities for triangular meshes obtained from regular surfaces. First, we introduce a new notation of the local tangential polygon and lift functions and vector fields on a triangular mesh to the local tangential polygon. Then we use the centroid weights proposed by Chen and Wu [4] to define the discrete gradient of a function on a triangular mesh. We also use our new method to define the discrete Laplacian operator acting on functions on triangular meshes. Higher order differential operators can also be computed successively. Our approach is conceptually simple and easy to compute. Indeed, our LTL method also provides a unified algorithm to estimate the shape operator and curvatures of a triangular mesh and derivatives of functions and vector fields. We also compare three different methods : our method, the least square method and Akima’s method to compute the gradients of functions.     

Topology optimization of structures with gradient elastic material

Topology optimization of structures and mechanisms with microstructural length-scale effect is investigated based on gradient elasticity theory. To meet the higher-order continuity requirement in gradient elasticity theory, Hermite finite elements are used in the finite element implementation. As an alternative to the gradient elasticity, the staggered gradient elasticity that requires C ⁰-continuity, is also presented. The solid isotropic material with penalization (SIMP) like material interpolation schemes are adopted to connect the element density with the constitutive parameters of the gradient elastic solid. The effectiveness of the proposed formulations is demonstrated via numerical examples, where remarkable length-scale effects can be found in the optimized topologies of gradient elastic solids as compared with linear elastic solids.     

Partwise cross-parameterization via nonregular convex hull domains

In this paper, we propose a novel partwise framework for cross-parameterization between 3D mesh models. Unlike most existing methods that use regular parameterization domains, our framework uses nonregular approximation domains to build the cross-parameterization. Once the nonregular approximation domains are constructed for 3D models, different (and complex) input shapes are transformed into similar (and simple) shapes, thus facilitating the cross-parameterization process. Specifically, a novel nonregular domain, the convex hull, is adopted to build shape correspondence. We first construct convex hulls for each part of the segmented model, and then adopt our convex-hull cross-parameterization method to generate compatible meshes. Our method exploits properties of the convex hull, e.g., good approximation ability and linear convex representation for interior vertices. After building an initial cross-parameterization via convex-hull domains, we use compatible remeshing algorithms to achieve an accurate approximation of the target geometry and to ensure a complete surface matching. Experimental results show that the compatible meshes constructed are well suited for shape blending and other geometric applications.