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Dynamic Substructuring
Dynamic Substructuring (DS) is an engineering tool used to Physical model, model and Analysis, analyse the Dynamics (mechanics), dynamics of mechanical systems by means of its components or substructures. Using the dynamic substructuring approach one is able to analyse the dynamic behaviour of substructures separately and to later on calculate the assembled dynamics using coupling procedures. Dynamic substructuring has several advantages over the analysis of the fully assembled system: * Substructures can be modelled in the domain that is most appropriate, e.g. experimentally obtained substructures can be combined with Finite element method in structural mechanics, numerical models. * Large and/or complex systems can be optimized on substructure level. * Numerical computation load can be reduced as solving several substructures is computationally less demanding than solving one large system. * Substructure models of different development groups can be shared and combined without expo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more specialized List of engineering branches, fields of engineering, each with a more specific emphasis on particular areas of applied mathematics, applied science, and types of application. See glossary of engineering. The term ''engineering'' is derived from the Latin ''ingenium'', meaning "cleverness" and ''ingeniare'', meaning "to contrive, devise". Definition The American Engineers' Council for Professional Development (ECPD, the predecessor of Accreditation Board for Engineering and Technology, ABET) has defined "engineering" as: The creative application of scientific principles to design or develop structures, machines, apparatus, or manufacturing processes, or works utilizing them singly or in combination; or to construct o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
High Tech
High technology (high tech), also known as advanced technology (advanced tech) or exotechnology, is technology that is at the cutting edge: the highest form of technology available. It can be defined as either the most complex or the newest technology on the market. The opposite of high tech is ''low technology'', referring to simple, often traditional or mechanical technology; for example, a slide rule is a low-tech calculating device. When high tech becomes old, it becomes low tech, for example vacuum tube electronics. The phrase was used in a 1958 ''The New York Times'' story advocating " atomic energy" for Europe: "... Western Europe, with its dense population and its high technology ...." Robert Metz used the term in a financial column in 1969, saying Arthur H. Collins of Collins Radio "controls a score of high technology patents in a variety of fields." and in a 1971 article used the abbreviated form, "high tech." A widely used classification of high-technological manu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Computing
Parallel computing is a type of computation in which many calculations or processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. There are several different forms of parallel computing: bit-level, instruction-level, data, and task parallelism. Parallelism has long been employed in high-performance computing, but has gained broader interest due to the physical constraints preventing frequency scaling.S.V. Adve ''et al.'' (November 2008)"Parallel Computing Research at Illinois: The UPCRC Agenda" (PDF). Parallel@Illinois, University of Illinois at Urbana-Champaign. "The main techniques for these performance benefits—increased clock frequency and smarter but increasingly complex architectures—are now hitting the so-called power wall. The computer industry has accepted that future performance increases must largely come from increasing the number of processors (or cores) on a die, rather tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Gradient Method
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems. The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization. It is commonly attributed to Magnus Hestenes and Eduard Stiefel, who programmed it on the Z4, and extensively researched it. The biconjugate gradient method provides a generalization to non-symmetric matrices. Various nonlinear conjugate gradient methods seek minima of nonlinear optimization problems. Description of the problem address ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Element Method
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, electromagnetics (where the technique is known as method of moments or abbreviated as MoM), fracture mechanics, and contact mechanics. Mathematical basis The integral equation may be regarded as an exact solution of the governing partial differential equation. The boundary element method attempts to use the given boundary conditions to fit boundary values into the integral equation, rather than values throughout the space defined by a partial differential equation. Once this is done, in the post-processing stage, the integral equation can then be used again to calculate numerically the solution directly at any desired point in the interior of the solution domain. BEM is applicable to problems for which Green's functions can be calculated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rayleigh Quotient
In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as: R(M,x) = . For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose x^ to the usual transpose x'. Note that R(M, c x) = R(M,x) for any non-zero scalar ''c''. Recall that a Hermitian (or real symmetric) matrix is diagonalizable with only real eigenvalues. It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value \lambda_\min (the smallest eigenvalue of ''M'') when ''x'' is v_\min (the corresponding eigenvector). Similarly, R(M, x) \leq \lambda_\max and R(M, v_\max) = \lambda_\max. The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms (such as Rayleigh quotient iteration) to obtain an eigenvalue approximation from an eigenvector approximation. The range of the Rayleigh quoti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rayleigh–Ritz Method
The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz. The name Rayleigh–Ritz is being debated vs. the Ritz method after Walther Ritz, since the numerical procedure has been published by Walther Ritz in 1908-1909. According to, Lord Rayleigh wrote a paper congratulating Ritz on his work in 1911, but stating that he himself had used Ritz's method in many places in his book and in another publication. This statement, although later disputed, and the fact that the method in the trivial case of a single vector results in the Rayleigh quotient make the arguable misnomer persist. According to, citing Richard Courant, both Lord Rayleigh and Walther Ritz independently conceived the idea of utilizing the equivalence between boundary value problems of partial differential equations on the one hand and problems of the calculus of variations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ritz Method
The Ritz method is a direct method to find an approximate solution for boundary value problems. The method is named after Walther Ritz, and is also commonly called the Rayleigh–Ritz method and the Ritz-Galerkin method. In quantum mechanics, a system of particles can be described in terms of an "energy functional" or Hamiltonian, which will measure the energy of any proposed configuration of said particles. It turns out that certain privileged configurations are more likely than other configurations, and this has to do with the eigenanalysis ("analysis of characteristics") of this Hamiltonian system. Because it is often impossible to analyze all of the infinite configurations of particles to find the one with the least amount of energy, it becomes essential to be able to approximate this Hamiltonian in some way for the purpose of numerical computations. The Ritz method can be used to achieve this goal. In the language of mathematics, it is exactly the finite element method use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable (creating a dichotomy for modeling purposes, as in binary classification). Discretization is also related to discrete mathematics, and is an important component of granular computing. In this context, ''discretization'' may also refer to modification of variable or category ''granularity'', as when multiple discrete variables are aggregated or multiple discrete categories fused. Whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce the amount to a level c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed-form Expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration. The set of operations and functions may vary with author and context. Example: roots of polynomials The solutions of any quadratic equation with complex coefficients can be expressed in closed form in terms of addition, subtraction, multiplication, division, and square root extraction, each of which is an elementary function. For example, the quadratic equation :ax^2+bx+c=0, is tractable since its solutions can be expressed as a closed-form expression, i.e. in terms of elementary functions: :x=\frac. Similarly, solutions of cubic and quartic (third and fourth degree) equations can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Schwarz
Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis. Life Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). In 1868 he married Marie Kummer, who was the daughter to the mathematician Ernst Eduard Kummer and Ottilie née Mendelssohn (a daughter of Nathan Mendelssohn's and granddaughter of Moses Mendelssohn). Schwarz and Kummer had six children, including his daughter Emily Schwarz. Schwarz originally studied chemistry in Berlin but Ernst Eduard Kummer and Karl Theodor Wilhelm Weierstrass persuaded him to change to mathematics. He received his Ph.D. from the Universität Berlin in 1864 and was advised by Kummer and Weierstrass. Between 1867 and 1869 he worked at the University of Halle, then at the Swiss Federal Polytechnic. From 1875 he worked at Göttingen University, dealing with the subjects of complex analysis, differential geometry and the calculus of variations. He ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
