scholarly journals Simulating deformable objects for computer animation: A numerical perspective

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Uri M. Ascher ◽  
Egor Larionov ◽  
Seung Heon Sheen ◽  
Dinesh K. Pai
Keyword(s):  
Numerical Analysis ◽  
Dynamical Systems ◽  
Computer Animation ◽  
Applied Mathematics ◽  
A Priori ◽  
Numerical Algorithms ◽  
Complex Geometries ◽  
Deformable Objects ◽  
Highly Oscillatory ◽  
The Mathematical Model

<p style='text-indent:20px;'>We examine a variety of numerical methods that arise when considering dynamical systems in the context of physics-based simulations of deformable objects. Such problems arise in various applications, including animation, robotics, control and fabrication. The goals and merits of suitable numerical algorithms for these applications are different from those of typical numerical analysis research in dynamical systems. Here the mathematical model is not fixed <i>a priori</i> but must be adjusted as necessary to capture the desired behaviour, with an emphasis on effectively producing lively animations of objects with complex geometries. Results are often judged by how realistic they appear to observers (by the "eye-norm") as well as by the efficacy of the numerical procedures employed. And yet, we show that with an adjusted view numerical analysis and applied mathematics can contribute significantly to the development of appropriate methods and their analysis in a variety of areas including finite element methods, stiff and highly oscillatory ODEs, model reduction, and constrained optimization.</p>

4. Do you know the way to solve equations? Spinning tops and chaotic rabbits

2018 ◽  
pp. 48-69
Author(s):  
Alain Goriely
Keyword(s):  
Numerical Analysis ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Numerical Solution ◽  
Applied Mathematics ◽  
Scientific Models ◽  
Henri Poincaré ◽  
Principal Tool ◽  
The Difference ◽  
The Way

In applied mathematics it is of the greatest importance to solve equations. These solutions provide information on key quantities and allow us to give specific answers to scientific problems. ‘Do you know the way to solve equations? Spinning tops and chaotic rabbits’ describes the ways to solve equations and differential equations, outlining the key work of mathematicians Sofia Kovalevskaya, Pierre-Simon Laplace, Paul Painlevé, and Henri Poincaré, whose discovery led to the birth of the theory of chaos and dynamical systems. The difference between an exact and a numerical solution is also explained. Numerical analysis has become the principal tool for querying and solving scientific models.

Introduction to Eigenanalysis

2011 ◽  
Author(s):  
Gidon Eshel
Keyword(s):  
Data Analysis ◽  
Numerical Analysis ◽  
Dynamical Systems ◽  
Applied Mathematics ◽  
Algebraic Operation ◽  
Systems Modeling ◽  
Dynamical Systems Modeling ◽  
Algebraic Operations ◽  
Eigen Decomposition ◽  
Data Matrices

Eigenanalysis and its numerous offsprings form the suite of algebraic operations most important and relevant to data analysis, as well as to dynamical systems, modeling, numerical analysis, and related key branches of applied mathematics. This chapter introduces and places in a broader context the algebraic operation of eigen-decomposition. To have eigen-decomposition, a matrix must be square. Yet data matrices are very rarely square. The direct relevance of eigen-decomposition to data analysis is therefore limited. Indirectly, however, generalized eigenanalysis is enormously important to studying data matrices. Because of the centrality of generalized eigenanalysis to data matrices, and because those generalizations build, algebraically and logically, on eigenanalysis itself, it makes sense to discuss eigenanalysis at some length.

Noninvertible Piecewise Linear Maps Applied to Chaos Synchronization and Secure Communications

1997 ◽  
Vol 07 (07) ◽  
pp. 1617-1634 ◽  
Author(s):  
G. Millerioux ◽  
C. Mira
Keyword(s):  
Dynamical Systems ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Piecewise Linear ◽  
A Priori ◽  
Secure Communications ◽  
Technological Parameters ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Necessary And Sufficient

Recently, it was demonstrated that two chaotic dynamical systems can synchronize each other, leading to interesting applications as secure communications. We propose in this paper a special class of dynamical systems, noninvertible discrete piecewise linear, emphasizing on interesting advantages they present compared with continuous and differentiable nonlinear ones. The generic aspect of such systems, the simplicity of numerical implementation, and the robustness to mismatch of technological parameters make them good candidates. The classical concept of controllability in the control theory is presented and used in order to choose and predict the number of appropriate variables to be transmitted for synchronization. A necessary and sufficient condition of chaotic synchronization is established without computing numerical quantities, introducing a state affinity structure of chaotic systems which provides an a priori establishment of synchronization.

Numerical Analysis: Pure or Applied Mathematics?

Science ◽  
1965 ◽  
Vol 149 (3688) ◽  
pp. 1049-1050
Author(s):  
Jerome Rothstein
Keyword(s):  
Numerical Analysis ◽  
Applied Mathematics

scholarly journals An integral equation formulation of the N-body dielectric spheres problem. Part 1: Numerical analysis

2020 ◽  
Author(s):  
Muhammad Hassan ◽  
Benjamin Stamm
Keyword(s):  
Integral Equation ◽  
Numerical Analysis ◽  
Error Analysis ◽  
Convergence Rates ◽  
A Priori ◽  
Approximation Error ◽  
Spherical Particles ◽  
Integral Equation Formulation ◽  
Dielectric Spheres ◽  
The Stability

In this article, we analyse an integral equation of the second kind that represents the solution of N interacting dielectric spherical particles undergoing mutual polarisation. A traditional analysis can not quantify the scaling of the stability constants- and thus the approximation error- with respect to the number N of involved dielectric spheres. We develop a new a priori error analysis that demonstrates N-independent stability of the continuous and discrete formulations of the integral equation. Consequently, we obtain convergence rates that are independent of N.

scholarly journals Population Behavior in the Mathematical Model of the Spread of COVID19 Type SEIRS

2021 ◽  
Vol 6 (2) ◽  
pp. 83-88
Author(s):  
Asmaidi As Med ◽  
Resky Rusnanda
Keyword(s):  
Mathematical Modeling ◽  
Numerical Simulation ◽  
Mathematical Model ◽  
Everyday Life ◽  
Applied Mathematics ◽  
Living Things ◽  
Simulation Results ◽  
Parameter Values ◽  
The Mathematical Model ◽  
Disease Free

Mathematical modeling utilized to simplify real phenomena that occur in everyday life. Mathematical modeling is popular to modeling the case of the spread of disease in an area, the growth of living things, and social behavior in everyday life and so on. This type of research is included in the study of theoretical and applied mathematics. The research steps carried out include 1) constructing a mathematical model type SEIRS, 2) analysis on the SEIRS type mathematical model by using parameter values for conditions 1and , 3) Numerical simulation to see the behavior of the population in the model, and 4) to conclude the results of the numerical simulation of the SEIRS type mathematical model. The simulation results show that the model stabilized in disease free quilibrium for the condition  and stabilized in endemic equilibrium for the condition .

scholarly journals In search of sports biomechanics’ holy grail: Can athlete-specific optimum sports techniques be identified?

2019 ◽  
Author(s):  
Paul Stephen Glazier ◽  
Sina Mehdizadeh
Keyword(s):  
Dynamical Systems ◽  
Holy Grail ◽  
Optimum Technique ◽  
Dynamic Landscape ◽  
Extended Practice ◽  
Technical Modifications ◽  
Sports Skills ◽  
The Mathematical Model ◽  
Near Future ◽  
Sports Biomechanics

The development of methods that can identify athlete-specific optimum sports techniques—arguably the holy grail of sports biomechanics—is one of the greatest challenges for researchers in the field. This ‘perspectives article’ critically examines, from a dynamical systems theoretical standpoint, the claim that athlete-specific optimum sports techniques can be identified through biomechanical optimisation modelling. To identify athlete-specific optimum sports techniques, dynamical systems theory suggests that a representative set of organismic constraints, along with their non-linear characteristics, needs to be identified and incorporated into the mathematical model of the athlete. However, whether the athlete will be able to adopt, and reliably reproduce, his/her predicted optimum technique will largely be dependent on his/her intrinsic dynamics. If the attractor valley corresponding to the existing technique is deep, or if the attractor valleys corresponding to the existing technique and the predicted optimum technique are in different topographical regions of the dynamic landscape, technical modifications may be challenging or impossible to reliably implement even after extended practice. The attractor layout defining the intrinsic dynamics of the athlete, therefore, needs to be determined to establish the likelihood of the predicted optimum technique being reliably attainable by the athlete. Given the limited set of organismic constraints typically used in mathematical models of athletes, combined with the methodological challenges associated with mapping the attractor layout of an athlete, it seems unlikely that athlete-specific optimum sports techniques will be identifiable through biomechanical optimisation modelling for the majority of sports skills in the near future.

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