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Smoothed Finite Element Method
Smoothed finite element methods (S-FEM) are a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining meshfree methods with the finite element method. S-FEM are applicable to solid mechanics as well as fluid dynamics problems, although so far they have mainly been applied to the former. Description The essential idea in the S-FEM is to use a finite element mesh (in particular triangular mesh) to construct numerical models of good performance. This is achieved by modifying the compatible strain field, or construct a strain field using only the displacements, hoping a Galerkin model using the modified/constructed strain field can deliver some good properties. Such a modification/construction can be performed within elements but more often beyond the elements (meshfree concepts): bring in the information from the neighboring elements. Naturally, the strain field has to satisfy certain conditions, and the standard Gal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Beta FEM
Beta (, ; uppercase , lowercase , or cursive ; or ) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Ancient Greek, beta represented the voiced bilabial plosive . In Modern Greek, it represents the voiced bilabial fricative while in borrowed words is instead commonly transcribed as μπ. Letters that arose from beta include the Roman letter and the Cyrillic letters and . Name Like the names of most other Greek letters, the name of beta was adopted from the acrophonic name of the corresponding letter in Phoenician, which was the common Semitic word ('house', compare and ). In Greek, the name was , pronounced in Ancient Greek. It is spelled in modern monotonic orthography and pronounced . History The letter beta was derived from the Phoenician letter beth . The letter Β had the largest number of highly divergent local forms. Besides the standard form (either rounded or pointed, ), there were forms as varied as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Numerical Differential Equations
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Numerical may refer to: * Number * Numerical digit * Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Continuum Mechanics
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mechanics deals with ''deformable bodies'', as opposed to rigid bodies. A continuum model assumes that the substance of the object completely fills the space it occupies. While ignoring the fact that matter is made of atoms, this provides a sufficiently accurate description of matter on length scales much greater than that of inter-atomic distances. The concept of a continuous medium allows for intuitive analysis of bulk matter by using differential equations that describe the behavior of such matter according to physical laws, such as mass conservation, momentum conservation, and energy conservation. Information about the specific material is expressed in constitutive relationships. Continuum mechanics treats the physical properties of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Loubignac Iteration
In applied mathematics, Loubignac iteration is an iterative method in finite element methods. It gives continuous stress field A stress field is the distribution of internal forces in a body that balance a given set of external forces. Stress fields are widely used in fluid dynamics and materials science. Consider that one can picture the stress fields as the stress cre .... It is named after Gilles Loubignac, who published the method in 1977. References Loubignac's paper Continuum mechanics Finite element method Numerical differential equations Partial differential equations Structural analysis {{applied-math-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Weakened Weak Form
Weakened weak form (or W2 form) is used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solid mechanics as well as fluid dynamics problems. Description For simplicity we choose elasticity problems (2nd order PDE) for our discussion.Liu, G.R. 2nd edn: 2009 ''Mesh Free Methods'', CRC Press. 978-1-4200-8209-9 Our discussion is also most convenient in reference to the well-known weak formulation, weak and strong form. In a strong formulation for an approximate solution, we need to assume displacement functions that are 2nd order differentiable. In a weak formulation, we create linear and bilinear forms and then search for a particular function (an approximate solution) that satisfy the weak statement. The bilinear form uses gradient of the functions that has only 1st order differentiation. Therefore, the requirement on the continuity of assumed displacement functions is weaker ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Meshfree Methods
In the field of numerical analysis, meshfree methods are those that do not require connection between nodes of the simulation domain, i.e. a mesh, but are rather based on interaction of each node with all its neighbors. As a consequence, original extensive properties such as mass or kinetic energy are no longer assigned to mesh elements but rather to the single nodes. Meshfree methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computing time and programming effort. The absence of a mesh allows Lagrangian simulations, in which the nodes can move according to the velocity field. Motivation Numerical methods such as the finite difference method, finite-volume method, and finite element method were originally defined on meshes of data points. In such a mesh, each point has a fixed number of predefined neighbors, and this connectivity between neighbors can be used to define mathematical operators like the derivative. These operators ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Finite Element Method
Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. Computers are usually used to perform the calculations required. With high-speed supercomputers, better solutions can be achieved and are often required to solve the largest and most complex problems. FEM is a general numerical method for solving partial differential equations in two- or three-space variables (i.e., some boundary value problems). There are also studies about using FEM to solve high-dimensional problems. To solve a problem, FEM subdivides a large system into smaller, simpler parts called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Basis Function
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors. In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points). Examples Monomial basis for ''Cω'' The monomial basis for the vector space of analytic functions is given by \. This basis is used in Taylor series, amongst others. Monomial basis for polynomials The monomial basis also forms a basis for the vector space of polynomials. After all, every polynomial can be written as a_0 + a_1x^1 + a_2x^2 + \cdots + a_n x^n for some n \in \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; that is, a function of class C^k is a function that has a th derivative that is continuous in its domain. A function of class C^\infty or C^\infty-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous). Generally, the term smooth function refers to a C^-function. However, it may also mean "sufficiently differentiable" for the problem under consideration. Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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B-spline
In numerical analysis, a B-spline (short for basis spline) is a type of Spline (mathematics), spline function designed to have minimal Support (mathematics), support (overlap) for a given Degree of a polynomial, degree, smoothness, and set of breakpoints (Knot (mathematics), knots that partition its Domain of a function, domain), making it a fundamental building block for all spline functions of that degree. A B-spline is defined as a piecewise polynomial of Order (mathematics), order n, meaning a degree of n - 1. It’s built from sections that meet at these knots, where the continuity of the function and its Derivative, derivatives depends on how often each knot repeats (its multiplicity). Any spline function of a specific degree can be uniquely expressed as a linear combination of B-splines of that degree over the same knots, a property that makes them versatile in mathematical modeling. A special subtype, cardinal B-splines, uses equidistant knots. The concept of B-splines tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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International Journal For Numerical Methods In Biomedical Engineering
''International Journal for Numerical Methods in Biomedical Engineering'' is a peer-reviewed scientific journal co-published monthly by Wiley. Established in 1985, it covers fundamentals and applications of numerical modeling in biomedical engineering. Its editor-in-chief is Perumal Nithiarasu (Swansea University). Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2023 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 2.2. References External links *{{official website, url=onlinelibrary.wiley.com/loi/20407947 Monthly journals Wiley (publisher) academic journals Academic journals established in 1985 English-language journals Biomedical engineering journals Computational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |