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Model Order Reduction
Model order reduction (MOR) is a technique for reducing the computational complexity of mathematical models in numerical simulations. As such it is closely related to the concept of metamodeling, with applications in all areas of mathematical modelling. Overview Many modern mathematical models of real-life processes pose challenges when used in numerical simulations, due to complexity and large size (dimension). Model order reduction aims to lower the computational complexity of such problems, for example, in simulations of large-scale dynamical systems and control systems. By a reduction of the model's associated state space dimension or degrees of freedom, an approximation to the original model is computed which is commonly referred to as a reduced order model. Reduced order models are useful in settings where it is often unfeasible to perform numerical simulations using the complete full order model. This can be due to limitations in computational resources or the requiremen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross Gramian
In control theory, the cross Gramian (W_X, also referred to by W_) is a Gramian matrix used to determine how controllable and observable a linear system is. For the stable time-invariant linear system :\dot = A x + B u \, :y = C x \, the cross Gramian is defined as: :W_X := \int_0^\infty e^ BC e^ dt \, and thus also given by the solution to the Sylvester equation: :A W_X + W_X A = -BC \, This means the cross Gramian is not strictly a Gramian matrix, since it is generally neither positive semi-definite nor symmetric. The triple (A,B,C) is controllable and observable, and hence minimal, if and only if the matrix W_X is nonsingular, (i.e. W_X has full rank, for any t > 0). If the associated system (A,B,C) is furthermore symmetric, such that there exists a transformation J with :AJ = JA^T \, :B = JC^T \, then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values: :, \lambda(W_X), = \sqrt. \, Thus the direct truncation of the Eigendeco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Dynamics F-16 Fighting Falcon
The General Dynamics F-16 Fighting Falcon is a single-engine Multirole combat aircraft, multirole fighter aircraft originally developed by General Dynamics for the United States Air Force (USAF). Designed as an air superiority day fighter, it evolved into a successful night fighter, all-weather multirole aircraft. Over 4,600 aircraft have been built since production was approved in 1976. Although no longer being purchased by the U.S. Air Force, improved versions are being built for export customers. In 1993, General Dynamics sold its aircraft manufacturing business to the Lockheed Corporation, which in turn became part of Lockheed Martin after a 1995 merger with Martin Marietta. The Fighting Falcon's key features include a frameless bubble canopy for good visibility, side-stick, side-mounted control stick to ease control while maneuvering, an ejection seat reclined 30 degrees from vertical to reduce the effect of g-forces on the Aircraft pilot, pilot, and the first use of a rel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aeronautics
Aeronautics is the science or art involved with the study, design, and manufacturing of air flight–capable machines, and the techniques of operating aircraft and rockets within the atmosphere. The British Royal Aeronautical Society identifies the aspects of "aeronautical Art, Science and Engineering" and "The profession of Aeronautics (which expression includes Astronautics)." While the term originally referred solely to ''operating'' the aircraft, it has since been expanded to include technology, business, and other aspects related to aircraft. The term "aviation" is sometimes used interchangeably with aeronautics, although "aeronautics" includes lighter-than-air craft such as airships, and includes ballistic vehicles while "aviation" technically does not. A significant part of aeronautical science is a branch of dynamics called aerodynamics, which deals with the motion of air and the way that it interacts with objects in motion, such as an aircraft. History Early ideas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Turbulent Structure
Turbulent flows are complex multi-scale and chaotic motions that need to be classified into more elementary components, referred to coherent turbulent structures. Such a structure must have temporal coherence, i.e. it must persist in its form for long enough periods that the methods of time-averaged statistics can be applied. Coherent structures are typically studied on very large scales, but can be broken down into more elementary structures with coherent properties of their own, such examples include hairpin vortices. Hairpins and coherent structures have been studied and noticed in data since the 1930s, and have been since cited in thousands of scientific papers and reviews.#Green, Sheldon I., “Fluid Vortices: Fluid mechanics and its applications” Dordrecht: Kluwer Academic Publishers, 1995. Print. https://books.google.com/books?id=j6qE7YAwwCoC&pg=PA254&lpg=PA254&dq=theodorsen+1952+hairpin&source=bl&ots=S9f7BlMhkg&sig=0qx5dJdvceQf22gm0li0Rt7UtL4&hl=en&sa=X&ei=1gNcU8DyOJWuyASB ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers. Turbulence is commonly observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason turbulence is commonly realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases. This increases the energy needed to pump fluid through a pipe. The onset of turbulence can be predicted by the dimensionless Rey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Navier–Stokes Equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes). The Navier–Stokes equations mathematically express conservation of momentum and conservation of mass for Newtonian fluids. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing ''viscous flow''. The difference between them and the closely related Euler equations is that Navier–Stokes equations take ... [...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] |
Boltzmann Equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3. The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number. The equation arises not by analyzing the individual positions and momenta of each particle in the fluid but rather by considering a probability distribution for the positio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structural Mechanics
Structural mechanics or Mechanics of structures is the computation of deformations, deflections, and internal forces or stresses (''stress equivalents'') within structures, either for design or for performance evaluation of existing structures. It is one subset of structural analysis. Structural mechanics analysis needs input data such as structural loads, the structure's geometric representation and support conditions, and the materials' properties. Output quantities may include support reactions, stresses and displacements. Advanced structural mechanics may include the effects of stability and non-linear behaviors. Mechanics of structures is a field of study within applied mechanics that investigates the behavior of structures under mechanical loads, such as bending of a beam, buckling of a column, torsion of a shaft, deflection of a thin shell, and vibration of a bridge. There are three approaches to the analysis: the energy methods, flexibility method or direct stiffness ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation. Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
