Highorder mimetic finite difference methods on nonuniform grids j. Such discretizations have been found useful in continuum mechanics and in electromagnetism. Charles fefferman and the clay mathematics institute technology. The goal is to construct a framework that is free of special assumptions about the type of material, the size of deformations, the geometry of the problem and so forth. Find materials for this course in the pages linked along the left. If the inline pdf is not rendering correctly, you can download the pdf file here. Consistent and mimetic discretizations in general relativity article pdf available in journal of mathematical physics 463 may 2004 with 40 reads how we measure reads. The family ofmimetic discretizations contains the classical mixed nite element discretizations on tetrahedral and hexahedral meshes 17 and the symmetric.
Compatible discretizations transform partial differential equations to discrete algebraic problems that mimic fundamental properties of the continuum equations. Mimetic discretizations for maxwells equations sciencedirect. It is a black box approach with the goal of predicting mechanical behavior in the absence of understanding for engineering and. The french mathematician augustinlouis cauchy was the first to formulate such models in the 19th century. In this paper, we explore the numerical approximation of discrete differential operators on nonuniform grids. Mimetic spatial discretizations have been used extensively to create simulation programs for problems in continuum mechanics, see 31 and the volume 27 in which this work appeared.
The fundamental assumption inscribed in the name is that materials are to be homogeneousassumed, isotropic, continuous and independent of any particular. Lecture notes applications of continuum mechanics to earth. V erani mox, dipartimento di matematica, politecnico di milano, piazza leonardo da v. The voronoi cell and the notion of natural neighbors are used to approximate the laplacian and. Saccomandi encyclopedia of life support systems eolss continuum mechanics is the branch of mechanics that studies deformable materials solids and fluids. Personal information postdoctoral fellow with bob russell and manfred trummer at the department of mathematics of simon fraser university. The orthogonal decomposition theorems for mimetic finite. Mimetic finite difference discretizations on triangular grids. The word mimetic comes from the greek word mimesis, the act of imitation. These methods mimic many fundamental properties of the underlying physical problem including conservation laws, symmetries in the solution, and the. This study derives geometric, variational discretizations of continuum theories arising in fluid dy.
Accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus provide reliable finite difference methods for approximating the solutions to a wide class of partial differential equations. There are numerous books on continuum mechanics with the main focus on the macroscale mechanical behavior of materials. We provide a common framework for mimetic discretizations using algebraic. Mimetic discretizations for the approximation of differential problems p. In addition, an effort has been made to correct numerous typographical errors that appeared in the first edition. Muskhelishvili 16th february ig6i published by the society for industrial and applied mathematics under a grantinaid from the national science foundation philadelphia, pennsylvania 1961. We have recently introduced a new technique for discretizing. It is a wellwritten mathematical introduction to classical continuum mechanics and deals with concepts such as elasticity, plasticity, viscoelasticity and viscoplasticity in nonlinear materials. Advancing the mimetic spectral element method tu delft. Continuum mechanics introduction to continuum mechanics j.
Mimetic discretizations of continuum mechanics this page provides information about mimetic discretizations of continuum mechanics problems, including problems in fluid mechanics, solid mechanics and electrodynamics. The family of mimetic discretizations contains the classical mixed. Nicolas robidouxs home page i maintain the mimetic discretizations bibliography homepage. Many algorithms used for a numerical simulation of physical problems solve discrete approximations of partial differential equations pdes. Continuum mechanics is the application of classical mechanics to continous media. Continuum mechanics english edition contributions in honor of the seventieth birthday of academician n. With that in mind, this introductory treatment of the principles of continuum mechanics is written as a text suitable for a. The mimetic theory of literary criticism places primary importance on how well a literary work imitates life. Jog cambridge university press 97811070951 continuum mechanics.
In numerical mathematics, the gradient discretisation method gdm is a framework which contains classical and recent numerical schemes for diffusion problems of various kinds. Compatible discretizations transform partial differential equations to discrete algebraic problems that mimic fundamental properties of the continuum equa tions. Mimetic discretisation techniques are a growing field in. Mimetic finite di erence methods an introduction andrea cangiani. The subject of all studies in continuum mechanics, and the domain of all physical quantities, is the material body. Consistent and mimetic discretizations in general relativity. A discrete vector calculus in tensor grids nicolas robidoux stanly steinberg abstract mimetic discretization methods for the numerical solution of continuum mechanics problems directly use vector calculus and di.
Mimetic theory of literary criticism pen and the pad. The mimetic discretization of the laplacian is given by the composition. We provide a common framework for mimetic discretizations using algebraic topology to guide our analysis. Continuum mechanics is the fundamental basis upon which several graduate courses in engineering science such as elasticity, plasticity, viscoelasticity, and. The goal is to construct a framework that is free of special assumptions about the type of material, the size of deformations, the. Continuum mechanics is a mathematical framework for studying the transmission of force through and deformation of materials of all types. Those talks were aimed at advanced graduate students, postdoctoral scholars, and faculty colleagues. Principles of mimetic discretizations of differential. A discretization of a continuum theory with constraints or conserved quantities is called mimetic if it mirrors the conserved laws or constraints of the. University of naples federico ii italy department of structural engineering continuum mechanics on manifolds giovanni romano with the collaboration of. Mimetic discretization methods for the numerical solution of continuum mechanics problems directly use vector calculus and differential forms. Unesco eolss sample chapters continuum mechanics introduction to continuum mechanics j.
A finite element framework for some mimetic finite. Saccomandi encyclopedia of life support systems eolss yf. More recently mimetic methods have been used in magnetohydrody. The mimetic finite di erence discretization of di usion. I am also one of the organizers of the nsf funded workshop on mimetic discretizations of continuum mechanics. Workshop on mimetic discretizations of continuum mechanics. Arnold, mathematical methods of classical mechanics, springerverlag, 1989. By discretizing the continuum theory, we mean that mimetic methods initially construct a discrete mathematical analog of a relevant description of continuum mechanics. Pdf consistent and mimetic discretizations in general. Lecture notes applications of continuum mechanics to. The schemes may be conforming or nonconforming, and may rely on very general polygonal or polyhedral meshes or. This provides a simple and transparent way to analyze such mimetic finite difference discretizations using the wellknown results from finite element theory.
The aim of a general theory of material behaviour is to provide a classified range of possibilities from which a user can select the constitutive. A divide and conquer strategy for featurepreserving discretizations. A finite element framework for some mimetic finite difference. The mimetic finite difference discretization of diffusion problem on. Mimetic discretizations compatible spatial discretizations finite element.
Siam journal on numerical analysis siam society for. Pdf mimetic discretizations of elliptic control problems. A material body b fxgis a compact measurable set of an in nite number of material elements x, called the material particles or material points, that can be placed in a onetoone correspondence with triplets of real numbers. Principles of mimetic discretizations of differential operators. Mimetic discretizations of continuum mechanics add. Continuum mechanics is the foundation for applied mechanics. A small amount of new material has been added in chapters 1, 3 and 4. Continuum mechanics cm is a beautiful and infinitely useful branch of mathematics, but the learning curve is relatively steep. We consider the following two model problems for u in a two dimensional simply connected domain 2. Unlike classical continuum mechanics books, this book summarizes the advances of continuum mechanics in several defined areas. The objective of the mimetic finite difference mfd method is to create discrete approximations that preserve important properties of continuum equations on general polygonal and polyhedral meshes. Stanly steinberg, university of new mexico navierstokes equations add. Numerical computation of discrete differential operators on. Continuum mechanics foundations and applications of mechanics volume i, third edition c.
Fully mimetic discretizations satisfy discrete analogs of. My appreciation for mechanics was nucleated by professors douglas amarasekara and munidasa ranaweera of the then university of ceylon, and was subsequently shaped and grew substantially under the in. Large sparse linear systems arising from mimetic discretization. A discretization of a continuum theory with constraints or conserved quantities is called mimetic if it mirrors the conserved laws or constraints of the continuum theory at the discrete level. Ima hot topics workshop compatible spatial discretizations for partial di erential equations, may 1115. The mimetic finite difference discretization has been successfully employed for solving problems of continuum mechanics 14, electromagnetics 8, gas. Numerical computation of discrete differential operators on nonuniform grids n. The idea for these lectures on continuum physics grew out of a short series of talks on materials physics at university of michigan, in the summer of 20. Mimetic discretization methods request pdf researchgate.
Partial differential equations mathematics archives www. However, most textbooks do not make for decent guides you need a competent professor. Continuum mechanics continuum mechanics and constitutive equations continuum mechanics pertains to the description of mechanical behavior of materials under the assumption that the material is a uniform continuum. Direct and conforming discretizations mimetic leastsquares principles for electromagnetics what are lsp and the reasons to use them. They have also been used to model inhomogeneous and anisotropic materials in two dimensions 22,21. Essays, links, references to the recent literature, people source. Typically, this description takes the form of a physical conservation or. Pdf principles of mimetic discretizations of differential operators. This electronic textbook is a revision to the textbook, introduction to continuum mechanics which was published by plenum press in 1989. In practice, mimetic critical theory often asks how well the literary work conveys universal truths and teaches the reader positive. Steinberg abstract by combining the supportoperators method with the mapping method, we have derived new mimetic fourth order accurate discretizations of the divergence, gradient, and laplacian on nonuniform grids. We have recently introduced a new technique for discretizing constrained theories. In both cases, we define discrete analogs of the continuum inner products 2. Mass spring vs continuum mechanics mass spring systems require.
1604 1375 121 91 1570 1558 128 6 1561 420 1507 1655 564 391 430 1059 776 1192 1213 316 249 1168 209 548 1187 1306 329 1420 928 518 1323 812 317 1107 551 1157 913 562 137 393 522 418 61