Semi-discrete finite difference multiscale scheme for a concrete corrosion model: approximation estimates and convergence

We propose a semi-discrete finite difference multiscale scheme for a concrete corrosion model consisting of a system of two-scale reaction-diffusion equations coupled with an ode. We prove energy and regularity estimates and use them to get the necessary compactness of the approximation estimates. Finally, we illustrate numerically the behavior of the two-scale finite difference approximation of the weak solution.Comment: 22 pages, 1 figure, submitted to Japan Journal of Industrial and Applied Mathematic

Mesh Deformation with Penalty-Based Collision Response

In this short note I consider the problem of triangular mesh deformation in R3 with simultaneous collision response between the deformed mesh and other fixed objects in the scene. My aim is an interactive mesh modeling system with quasi-static mesh deformation where the user manipulates a set of handles and the system helps him (interactively or in a post-processing step) to avoid or reduce collisions. In recent years a number of physically plausible mesh deformation methods were introduced that are based on finding the deformed geometry as a (local) minimum of a specifically designed deformation energy (see, e.g., [1, 2, 7]). In this note I will focus mainly on As-Rigid-As-Possible (ARAP) mesh deformation [7] due to its simplicity, sufficiently plausible results and good performance for the intended application in anatomical modeling. In ARAP, a mesh deformation energy is defined in terms of the current and deformed mesh geometry. It is a non-linear energy that tries to keep the deformation locally as close as possible to rigid transformation while satisfying user constraints. These are typically given by a set of fixed vertices (that do not change their position during the deformation) and a set of handle vertices (that are manipulated interactively by the user and are not affected by the deformation). Here, I combine ARAP mesh deformation with a standard penalty-based approach to collision response. In penalty methods artificial forces are introduced into the system when a collision is detected. The force is proportional to the penetration depth and its magnitude and direction act to resolve the collision (see, e.g., [8] and references therein). A nice and successful combination of physically plausible energy-based mesh deformation with penalty-based response to collision was presented in [5]. In their approach, the authors introduce a penalty force in the form of a spring with zero rest length and anisotropic Young\u27s modulus. In this note, the ARAP deformation energy is modified by adding a penalty term that penalizes vertex positions that are inside or near other objects in the scene. It consists of a smooth scalar function that takes the distance from a vertex to another mesh as its parameter. The choice of this function gives flexibility in the desired behavior of the response to collision (e.g., response to collision after it occured or repulsion before the collision occurs). Note that this approach does not guarantee that all collisions are avoided or resolved. Also, self-collisions are not considered

Mesh Deformation with Penalty-Based Collision Response

In this short note I consider the problem of triangular mesh deformation in R3 with simultaneous collision response between the deformed mesh and other fixed objects in the scene. My aim is an interactive mesh modeling system with quasi-static mesh deformation where the user manipulates a set of handles and the system helps him (interactively or in a post-processing step) to avoid or reduce collisions. In recent years a number of physically plausible mesh deformation methods were introduced that are based on finding the deformed geometry as a (local) minimum of a specifically designed deformation energy (see, e.g., [1, 2, 7]). In this note I will focus mainly on As-Rigid-As-Possible (ARAP) mesh deformation [7] due to its simplicity, sufficiently plausible results and good performance for the intended application in anatomical modeling. In ARAP, a mesh deformation energy is defined in terms of the current and deformed mesh geometry. It is a non-linear energy that tries to keep the deformation locally as close as possible to rigid transformation while satisfying user constraints. These are typically given by a set of fixed vertices (that do not change their position during the deformation) and a set of handle vertices (that are manipulated interactively by the user and are not affected by the deformation). Here, I combine ARAP mesh deformation with a standard penalty-based approach to collision response. In penalty methods artificial forces are introduced into the system when a collision is detected. The force is proportional to the penetration depth and its magnitude and direction act to resolve the collision (see, e.g., [8] and references therein). A nice and successful combination of physically plausible energy-based mesh deformation with penalty-based response to collision was presented in [5]. In their approach, the authors introduce a penalty force in the form of a spring with zero rest length and anisotropic Young's modulus. In this note, the ARAP deformation energy is modified by adding a penalty term that penalizes vertex positions that are inside or near other objects in the scene. It consists of a smooth scalar function that takes the distance from a vertex to another mesh as its parameter. The choice of this function gives flexibility in the desired behavior of the response to collision (e.g., response to collision after it occured or repulsion before the collision occurs). Note that this approach does not guarantee that all collisions are avoided or resolved. Also, self-collisions are not considered

[034] Homogenization Method and Multiscale Modeling

本レクチャーノートシリーズは、文部科学省21COEプログラム「機能数理学の構築と展開」（H.15-19年度）において作成したCOE Lecture Notes の続刊である。今後、レクチャーノートは、文部科学省大学院教育改革支援プログラム「産業界が求める数学博士と新修士養成（H19-21年度）および、新しく採択された同グローバルCOEプログラム「マス・フォア・インダストリ教育研究拠点」（H.21-24年度）の推進において招聘する国内外の研究者による講義の講義録として出版するものであ

Semi-discrete finite difference multiscale scheme for a concrete corrosion model : approximation estimates and convergence

We propose a semi-discrete finite difference multiscale scheme for a concrete corrosion model consisting of a system of two-scale reaction-diffusion equations coupled with an ode. We prove energy and regularity estimates and use them to get the necessary compactness of the approximation estimates. Finally, we illustrate numerically the behavior of the two-scale finite difference approximation of the weak solution

International Conference CoMFoS15

This book focuses on mathematical theory and numerical simulation related to various aspects of continuum mechanics, such as fracture mechanics, elasticity, plasticity, pattern dynamics, inverse problems, optimal shape design, material design, and disaster estimation related to earthquakes. Because these problems have become more important in engineering and industry, further development of mathematical study of them is required for future applications. Leading researchers with profound knowledge of mathematical analysis from the fields of applied mathematics, physics, seismology, engineering, and industry provide the contents of this book. They help readers to understand that mathematical theory can be applied not only to different types of industry, but also to a broad range of industrial problems including materials, processes, and products