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Hp-FEM
hp-FEM is a general version of the finite element method (FEM), a numerical analysis, numerical method for solving partial differential equations based on Piecewise, piecewise-polynomial approximations that employs elements of variable size ''(h)'' and Degree of a polynomial, polynomial degree ''(p)''. The origins of hp-FEM date back to the pioneering work of Barna A. Szabó and Ivo Babuška who discovered that the finite element method converges ''exponentially fast'' when the mesh is refined using a suitable combination of h-refinements (dividing elements into smaller ones) and P-FEM, p-refinements (increasing their polynomial degree). The exponential convergence makes the method very attractive compared to most other finite element methods, which only converge with an algebraic rate. The exponential convergence of hp-FEM was not only predicted theoretically, but also observed by numerous independent researchers. Differences from standard FEM The hp-FEM differs from the sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Element Method
The 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. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are 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 numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain. The sim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Agros2D
Agros2D is an open-source code for numerical solutions of 2D coupled problems in technical disciplines. Its principal part is a user interface serving for complete preprocessing and postprocessing of the tasks (it contains sophisticated tools for building geometrical models and input of data, generators of meshes, tables of weak forms for the partial differential equations and tools for evaluating results and drawing graphs and maps). The processor is based on the library Hermes containing the most advanced numerical algorithms for monolithic and fully adaptive solution of systems of generally nonlinear and nonstationary partial differential equations (PDEs) based on hp-FEM (adaptive finite element method of higher order of accuracy). Both parts of the code are written in C++.Karban, P., Mach, F., Kůs, P., Pánek, D., Doležel, I.: Numerical solution of coupled problems using code Agros2D, Computing, 2013, Volume 95, Issue 1 Supplement, pp 381-408 Features * Coupled Fields - W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-FEM
p-FEM or the p-version of the finite element method is a numerical method for solving partial differential equations. It is a discretization strategy in which the finite element mesh is fixed and the polynomial degrees of elements are increased such that the lowest polynomial degree, denoted by p_, approaches infinity. This is in contrast with the "h-version" or "h-FEM", a widely used discretization strategy, in which the polynomial degrees of elements are fixed and the mesh is refined such that the diameter of the largest element, denoted by h_ approaches zero. It was demonstrated on the basis of a linear elastic fracture mechanics problem that sequences of finite element solutions based on the p-version converge faster than sequences based on the h-version by Szabó and Mehta in 1978. The theoretical foundations of the p-version were established in a paper published Babuška, Szabó and Katz in 1981 where it was shown that for a large class of problems the asymptotic rate of co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Piecewise definition is actually a way of expressing the function, rather than a characteristic of the function itself. A distinct, but related notion is that of a property holding piecewise for a function, used when the domain can be divided into intervals on which the property holds. Unlike for the notion above, this is actually a property of the function itself. A piecewise linear function (which happens to be also continuous) is depicted as an example. Notation and interpretation Piecewise functions can be defined using the common functional notation, where the body of the function is an array of functions and associated subdomains. These subdomains together must cover the whole domain; often it is also required that they are pair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see order of a polynomial (other)). For example, the polynomial 7x^2y^3 + 4x - 9, which can also be written as 7x^2y^3 + 4x^1y^0 - 9x^0y^0, has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (x+1)^2 - (x-1)^2, one can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barna Szabó
Barna A. Szabó (born September 21, 1935) is a Hungarian-American engineer and educator, noted for his contributions on the finite element method, particularly the conception and implementation of the p- and hp-versions of the Finite Element Method. He is a founding member and fellow of the United States Association for Computational Mechanics, an external member of the Hungarian Academy of Sciences and fellow of the St. Louis Academy of Sciences. Life Barna Aladár Szabó was born in Martonvásár, Hungary, the son of József and Gizella (Iványi) Szabó. After graduating from the Franciscan High School in Esztergom in 1954, he was admitted to the Faculty of Mining Engineering of the Technical University of Heavy Industry in Miskolc (now University of Miskolc). Following the failed Hungarian uprising in 1956 he emigrated to Canada, where he resumed his undergraduate studies at the University of Toronto, receiving his Bachelor of Science in 1960. In 1963 he began part-time st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ivo Babuška
Ivo M. Babuška (born March 22, 1926, in Prague) is a Czech-American mathematician, noted for his studies of the finite element method and the proof of the Babuška–Lax–Milgram theorem in partial differential equations. One of the celebrated result in the finite elements is the so-called Ladyzenskaja–Babuška–Brezzi (LBB) condition (also referred to in some literature as Banach–Necas–Babuška (BNB)), which provides sufficient conditions for a stable mixed formulation. The LBB condition has guided mathematicians and engineers to develop state-of-the-art formulations for many technologically important problems like Darcy flow, Stokes flow, incompressible Navier–Stokes, nearly incompressible elasticity. He is also well known for his work on adaptive methods and the '' p-''- and '' hp-''-versions of the finite element method. He also developed the mathematical framework for the partition of unity methods. Babuska was elected as a member to the National Academy of Engi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 at finding 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, 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 simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degrees Of Freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or other physical processes are possible. This relates to the philosophical concept to the extent that people may be considered to have as much freedom as they are physically able to exercise. Applications Statistics In statistics, degrees of freedom refers to the number of variables in a statistic calculation that can vary. It can be calculated by subtracting the number of estimated parameters from the total number of values in the sample. For example, a sample variance calculation based on n samples will have n-1 degrees of freedom, because sample variance is calculated using the sample mean as an estimate of the actual mean. Mathematics In mathematics, this notion is formalized as the dimension of a manifold or an algebraic variety. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grad Sin H
Grad or grads may refer to: Places * Grad (toponymy) (Cyrillic: Град) is a Slavic word meaning "town", "city", "castle" or "fortified settlement" that appears in numerous Slavic toponyms ;Specific places named Grad: * Grad, Visoko, Bosnia and Herzegovina * Grad, Delčevo, North Macedonia * Grąd, Podlaskie Voivodeship, Poland * Grąd, West Pomeranian Voivodeship, Poland * Grad, Cerklje na Gorenjskem, Slovenia * Municipality of Grad, Slovenia ** Grad, Grad, a village, the seat of the municipality People * Grad (surname) Arts, entertainment, and media * ''Grad'' (EP), a Cactus Jack EP * Grad, the dragon in ''Ral Grad'' Education *Grad, short for a graduate, one who has graduated from an education program, also known as an alumnus *Grad school, short for graduate school Geometry and measurement * Gradian, a unit of angular measurement * Gradient of a scalar field, a differential operator in mathematics * Grad, a small unit in tuning very close to the schisma, which it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
