Fast character modeling with sketch-based PDE surfaces

© 2020, The Author(s). Virtual characters are 3D geometric models of characters. They have a lot of applications in multimedia. In this paper, we propose a new physics-based deformation method and efficient character modelling framework for creation of detailed 3D virtual character models. Our proposed physics-based deformation method uses PDE surfaces. Here PDE is the abbreviation of Partial Differential Equation, and PDE surfaces are defined as sculpting force-driven shape representations of interpolation surfaces. Interpolation surfaces are obtained by interpolating key cross-section profile curves and the sculpting force-driven shape representation uses an analytical solution to a vector-valued partial differential equation involving sculpting forces to quickly obtain deformed shapes. Our proposed character modelling framework consists of global modeling and local modeling. The global modeling is also called model building, which is a process of creating a whole character model quickly with sketch-guided and template-based modeling techniques. The local modeling produces local details efficiently to improve the realism of the created character model with four shape manipulation techniques. The sketch-guided global modeling generates a character model from three different levels of sketched profile curves called primary, secondary and key cross-section curves in three orthographic views. The template-based global modeling obtains a new character model by deforming a template model to match the three different levels of profile curves. Four shape manipulation techniques for local modeling are investigated and integrated into the new modelling framework. They include: partial differential equation-based shape manipulation, generalized elliptic curve-driven shape manipulation, sketch assisted shape manipulation, and template-based shape manipulation. These new local modeling techniques have both global and local shape control functions and are efficient in local shape manipulation. The final character models are represented with a collection of surfaces, which are modeled with two types of geometric entities: generalized elliptic curves (GECs) and partial differential equation-based surfaces. Our experiments indicate that the proposed modeling approach can build detailed and realistic character models easily and quickly

Meshless Collocation Methods for the Numerical Solution of Elliptic Boundary Valued Problems and the Rotational Shallow Water Equations on the Sphere

This dissertation thesis has three main goals: 1) To explore the anatomy of meshless collocation approximation methods that have recently gained attention in the numerical analysis community; 2) Numerically demonstrate why the meshless collocation method should clearly become an attractive alternative to standard finite-element methods due to the simplicity of its implementation and its high-order convergence properties; 3) Propose a meshless collocation method for large scale computational geophysical fluid dynamics models. We provide numerical verification and validation of the meshless collocation scheme applied to the rotational shallow-water equations on the sphere and demonstrate computationally that the proposed model can compete with existing high performance methods for approximating the shallow-water equations such as the SEAM (spectral-element atmospheric model) developed at NCAR. A detailed analysis of the parallel implementation of the model, along with the introduction of parallel algorithmic routines for the high-performance simulation of the model will be given. We analyze the programming and computational aspects of the model using Fortran 90 and the message passing interface (mpi) library along with software and hardware specifications and performance tests. Details from many aspects of the implementation in regards to performance, optimization, and stabilization will be given. In order to verify the mathematical correctness of the algorithms presented and to validate the performance of the meshless collocation shallow-water model, we conclude the thesis with numerical experiments on some standardized test cases for the shallow-water equations on the sphere using the proposed method

Eine gitterfreie Raum-Zeit-Kollokationsmethode für gekoppelte Probleme auf Gebieten mit komplizierten Rändern

In der vorliegenden Arbeit wird eine neuartige gitterfreie Raum-Zeit-Kollokationsmethode (engl. STMCM) zur Lösung von Systemen partieller und gewöhnlicher Differentialgleichungen durch eine konsistente Diskretisierung in Raum und Zeit als Alternative zu den etablierten netzbasierten Verfahren vorgeschlagen. Die STMCM gehört zur Klasse der tatsächlich gitterfreien Methoden, die nur mit Punktwolken ohne a priori Netzkonnektivität arbeiten und kein Diskretisierungsnetz benötigen. Das Verfahren basiert auf der Interpolating Moving Least Squares Methode, die eine vereinfachte Erfüllung der Randbedingungen durch die von den Kernfunktionen erfüllte Kronecker-Delta-Eigenschaft ermöglicht, was beim größten Teil anderer netzfreier Verfahren nicht der Fall ist. Ein Regularisierungsverfahren zur Bewältigung des beim Aufbau der Kernfunktionen auftretenden Singularitätsproblems, sowie zur Berechnung aller benötigten Ableitungen der Kernfunktionen wird dargelegt. Ziel ist es dabei, eine Methode zu entwickeln, die die Einfachheit der Verfahren zur Lösung partieller Differentialgleichungen in starker Form mit den Vorteilen der gitterfreien Verfahren, insbesondere mit Blick auf gekoppelte Probleme des Ingenieurwesens mit sich bewegenden Grenzflächen, verknüpft. Die vorgeschlagene Methode wird zunächst zur Lösung linearer und nichtlinearer partieller sowie gewöhnlicher Differentialgleichungen angewendet. Dabei werden deren Konvergenz- und Genauigkeitseigenschaften untersucht. Die implizite Rekonstruktion der Gebiete mit komplizierten Rändern als Abbildungsstrategie zur Punktwolken-Streuung wird durch die Interpolation von Punktwolkendaten in zwei und drei Raumdimensionen demonstriert. Anhand der Modelle zur Simulation von Biofilm- und Tumor-Wachstumsprozessen werden Anwendungsbeispiele aus dem Bereich der Umweltwissenschaften und der Medizintechnik dargestellt.In this thesis an innovative Space-Time Meshfree Collocation Method (STMCM) for solving systems of nonlinear ordinary and partial differential equations by a consistent discretization in both space and time is proposed as an alternative to established mesh-based methods. The STMCM belongs to the class of truly meshfree methods, i.e. the methods which do not have any underlying mesh, but work on a set of nodes only without an a priori node-to-node connectivity. The STMCM is constructed using the Interpolating Moving Least Squares technique, allowing a simplified implementation of boundary conditions due to fulfilment of the Kronecker delta property by the kernel functions, which is not the case for the major part of other meshfree methods. A regularization technique to overcome the singularity-by-construction problem and compute all necessary derivatives of the kernel functions is presented. The goal is to design a method that combines the simplicity and straightforwardness of the strong-form computational techniques with the advantages of meshfree methods over the classical ones, especially for coupled engineering problems involving moving interfaces. The proposed STMCM is applied to linear and nonlinear partial and ordinary differential equations of different types and its accuracy and convergence properties are studied. The power of the technique is demonstrated by implicit reconstruction of domains with complex boundaries via interpolation of point cloud data in two and three space dimensions as a `mapping' strategy for distribution of computational points within such domains. Applications from the fields of environmental and medical engineering are presented by means of a mathematical model for simulating a biofilm growth and a nonlinear model of tumour growth processes

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Semi-Infinite Structure Analysis with Bimodular Materials with Infinite Element

The modulus of elasticity of some materials changes under tensile and compressive states is simulated by constructing a typical material nonlinearity in a numerical analysis in this paper. The meshless Finite Block Method (FBM) has been developed to deal with 3D semi-infinite structures in the bimodular materials in this paper. The Lagrange polynomial interpolation is utilized to construct the meshless shape function with the mapping technique to transform the irregular finite domain or semi-infinite physical solids into a normalized domain. A shear modulus strategy is developed to present the nonlinear characteristics of bimodular material. In order to verify the efficiency and accuracy of FBM, the numerical results are compared with both analytical and numerical solutions provided by Finite Element Method (FEM) in four examples

Accelerating the Performance of a Novel Meshless Method Based on Collocation With Radial Basis Functions By Employing a Graphical Processing Unit as a Parallel Coprocessor

In recent times, a variety of industries, applications and numerical methods including the meshless method have enjoyed a great deal of success by utilizing the graphical processing unit (GPU) as a parallel coprocessor. These benefits often include performance improvement over the previous implementations. Furthermore, applications running on graphics processors enjoy superior performance per dollar and performance per watt than implementations built exclusively on traditional central processing technologies. The GPU was originally designed for graphics acceleration but the modern GPU, known as the General Purpose Graphical Processing Unit (GPGPU) can be used for scientific and engineering calculations. The GPGPU consists of massively parallel array of integer and floating point processors. There are typically hundreds of processors per graphics card with dedicated high-speed memory. This work describes an application written by the author, titled GaussianRBF to show the implementation and results of a novel meshless method that in-cooperates the collocation of the Gaussian radial basis function by utilizing the GPU as a parallel co-processor. Key phases of the proposed meshless method have been executed on the GPU using the NVIDIA CUDA software development kit. Especially, the matrix fill and solution phases have been carried out on the GPU, along with some post processing. This approach resulted in a decreased processing time compared to similar algorithm implemented on the CPU while maintaining the same accuracy

A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces

In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded in $\mathbb{R}^3$, these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions (RBFs) and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at "scattered" locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented

Numerical Relativity: A review

Computer simulations are enabling researchers to investigate systems which are extremely difficult to handle analytically. In the particular case of General Relativity, numerical models have proved extremely valuable for investigations of strong field scenarios and been crucial to reveal unexpected phenomena. Considerable efforts are being spent to simulate astrophysically relevant simulations, understand different aspects of the theory and even provide insights in the search for a quantum theory of gravity. In the present article I review the present status of the field of Numerical Relativity, describe the techniques most commonly used and discuss open problems and (some) future prospects.Comment: 2 References added; 1 corrected. 67 pages. To appear in Classical and Quantum Gravity. (uses iopart.cls