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Proper Orthogonal Decomposition
The proper orthogonal decomposition is a numerical method that enables a reduction in the complexity of computer intensive simulations such as computational fluid dynamics and structural analysis (like crash simulations). Typically in fluid dynamics and turbulences analysis, it is used to replace the Navier–Stokes equations by simpler models to solve. It belongs to a class of algorithms called ''model order reduction'' (or in short ''model reduction''). What it essentially does is to train a model based on simulation data. To this extent, it can be associated with the field of machine learning. POD and PCA The main use of POD is to decompose a physical field (like pressure, temperature in fluid dynamics or stress and deformation in structural analysis), depending on the different variables that influence its physical behaviors. As its name hints, it's operating an Orthogonal Decomposition along with the Principal Components of the field. As such it is assimilated with the pr ... [...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] |
Eigenvalues And Eigenvectors
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a 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 numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of 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 as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Continuum Mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century. Explanation A continuum model assumes that the substance of the object fills the space it occupies. Modeling objects in this way ignores the fact that matter is made of atoms, and so is not continuous; however, on length scales much greater than that of inter-atomic distances, such models are highly accurate. These models can be used to derive differential equations that describe the behavior of such objects using physical laws, such as mass conservation, momentum conservation, and energy conservation, and some information about the material is provided by constitutive relationships. Continuum mechanics deals with the physical properties of solids and fluids which are independent of any particular coordinate sy ... [...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] |
Covariance Matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the x and y directions contain all of the necessary information; a 2 \times 2 matrix would be necessary to fully characterize the two-dimensional variation. The covariance matrix of a random vector \mathbf is typically denoted by \operatorname_ or \Sigma. Definition Throughout this article, boldfaced unsubsc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
POD Snapshots
Pod or POD may refer to: Biology * Pod (fruit), a type of fruit of a flowering plant * Husk or pod of a legume * Pod of whales or other marine mammals * "-pod", a suffix meaning "foot" used in taxonomy Electronics and computing * Proper orthogonal decomposition in the field of numerical simulation *Plain old data in computing, data distinct from an object * Plain Old Documentation, a documentation tool for the computer language Perl * Point of delivery (networking) * Pseudo open drain, an electronics interface technology * Personal online data stores, storage of personal data for the web decentralization project Solid * Pod, the basic scheduling unit in Kubernetes Film and television * ''Pod'' (film), an American horror film * Podracer, a type of vehicle from the Star Wars universe * Orthotube or pod, a fictional security device in ''Spooks'' * Pod, the growth medium for the replacements in ''Invasion of the Body Snatchers'' * Pod, a fictional organic gaming console featu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Value Decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an \ m \times n\ complex matrix is a factorization of the form \ \mathbf = \mathbf\ , where is an \ m \times m\ complex unitary matrix, \ \mathbf\ is an \ m \times n\ rectangular diagonal matrix with non-negative real numbers on the diagonal, is an n \times n complex unitary matrix, and \ \mathbf\ is the conjugate transpose of . Such decomposition always exists for any complex matrix. If is real, then and can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted \ \mathbf^\mathsf\ . The diagonal entries \ \sigma_i = \Sigma_\ of \ \mathbf\ are uniquely determined by and are known as the singular values of . The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Fluid Dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate the free-stream flow of the fluid, and the interaction of the fluid ( liquids and gases) with surfaces defined by boundary conditions. With high-speed supercomputers, better solutions can be achieved, and are often required to solve the largest and most complex problems. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial validation of such software is typically performed using experimental apparatus such as wind tunnels. In addition, previously performed analytical or empirical analysis of a particular problem can be used for comparison. A final validation is often performed using full-scale testing, such as flight tests. CFD is applied to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
