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Temporal Discretization
Temporal discretization is a mathematical technique applied to transient problems that occur in the fields of applied physics and engineering. Transient problems are often solved by conducting simulations using computer-aided engineering (CAE) packages, which require discretizing the governing equations in both space and time. Such problems are unsteady (e.g. flow problems), and therefore require solutions in which position varies as a function of time. Temporal discretization involves the integration of every term in different equations over a time step (\Delta t). The spatial domain can be discretized to produce a semi-discrete form: \frac(x,t) = F(\varphi).~ If the discretization is done using backward differences, the first-order temporal discretization is given as: \frac = F(\varphi), And the second-order discretization is given as: \frac = F(\varphi), where * \varphi is a scalar quantity. * n + 1 is the value at the next time level, t + \Delta t. * n is the value ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transient (civil Engineering)
In civil engineering, a transient is used to refer to any pressure wave that is short lived (i.e. not static pressure or pressure differential due to friction/minor loss in flow). The most common occurrence of this is called water hammer. In a pipe network, when a valve or pump is suddenly shut off, the water flowing in an adjacent pipe is suddenly forced to stop. A region of high pressure builds up immediately behind said valve or pump and a region of low pressure forms in front of it. The momentum of the water is suddenly transferred into the fitting and Newton's Third Law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ... kicks in forming a high-pressure region of water as it all "piles up" in the pipe. This high pressure region then travels back along the pipe in the form of a wav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical potential. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, like in spinodal decomposition. The concept of diffusion is widely used in many fields, including physics (particle diffusion), chemistry, biology, sociology, economics, and finance (diffusion of people, ideas, and price values). The central idea of diffusion, however, is common to all of these: a substance or collection undergoing diffusion spreads out from a point or location at which there is a higher concentration of that substance or collection. A gradient is the change in the value of a quantity, for example, concentration, pressure, or temperature with the change in another variable, usually distance. A change in c ... [...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] |
Von Neumann Stability Analysis
The term ''von'' () is used in German language surnames either as a nobiliary particle indicating a noble patrilineality, or as a simple preposition used by commoners that means ''of'' or ''from''. Nobility directories like the ''Almanach de Gotha'' often abbreviate the noble term ''von'' to ''v.'' In medieval or early modern names, the ''von'' particle was at times added to commoners' names; thus, ''Hans von Duisburg'' meant "Hans from he city ofDuisburg". This meaning is preserved in Swiss toponymic surnames and in the Dutch or Afrikaans '' van'', which is a cognate of ''von'' but does not indicate nobility. Usage Germany and Austria The abolition of the monarchies in Germany and Austria in 1919 meant that neither state has a privileged nobility, and both have exclusively republican governments. In Germany, this means that legally ''von'' simply became an ordinary part of the surnames of the people who used it. There are no longer any legal privileges or constraints asso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lax–Wendroff Method
The Lax–Wendroff method, named after Peter Lax and Burton Wendroff, is a numerical method for the solution of hyperbolic partial differential equations, based on finite difference A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...s. It is second-order accurate in both space and time. This method is an example of explicit time integration where the function that defines the governing equation is evaluated at the current time. Definition Suppose one has an equation of the following form: \frac + \frac = 0 where and are independent variables, and the initial state, is given. Linear case In the linear case, where , and is a constant, u_i^ = u_i^n - \frac A\left u_^ - u_^ \right+ \frac A^2\left u_^ -2 u_^ + u_^ \right Here n refers to the t dimension and i refers to the x di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Courant–Friedrichs–Lewy Condition
In mathematics, the convergence condition by Courant–Friedrichs–Lewy is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically. It arises in the numerical analysis of explicit time integration schemes, when these are used for the numerical solution. As a consequence, the time step must be less than a certain time in many explicit time-marching computer simulations, otherwise the simulation produces incorrect results. The condition is named after Richard Courant, Kurt Friedrichs, and Hans Lewy who described it in their 1928 paper. Heuristic description The principle behind the condition is that, for example, if a wave is moving across a discrete spatial grid and we want to compute its amplitude at discrete time steps of equal duration, then this duration must be less than the time for the wave to travel to adjacent grid points. As a corollary, when the grid point separation is reduced, the upper limit for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Current Source
A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it. A current source is the dual of a voltage source. The term ''current sink'' is sometimes used for sources fed from a negative voltage supply. Figure 1 shows the schematic symbol for an ideal current source driving a resistive load. There are two types. An ''independent current source'' (or sink) delivers a constant current. A ''dependent current source'' delivers a current which is proportional to some other voltage or current in the circuit. Background , - align="center" , style="padding: 1em 2em 0;", , style="padding: 1em 2em 0;", , - align="center" , Voltage source , Current source , - align="center" , style="padding: 1em 2em 0;", , style="padding: 1em 2em 0;", , - align="center" , Controlled voltage source , Controlled current source , - align="center" , style="padding: 1em 2em 0;", , style="padding: 1em 2em 0;", , - align ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convection is unspecified, convection due to the effects of thermal expansion and buoyancy can be assumed. Convection may also take place in soft solids or mixtures where particles can flow. Convective flow may be transient (such as when a multiphase mixture of oil and water separates) or steady state (see Convection cell). The convection may be due to gravitational, electromagnetic or fictitious body forces. Heat transfer by natural convection plays a role in the structure of Earth's atmosphere, its oceans, and its mantle. Discrete convective cells in the atmosphere can be identified by clouds, with stronger convection resulting in thunderstorms. Natural convection also plays a role in stellar physics. Convection is often categorised or d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crank–Nicolson Method
In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time. It is implicit in time, can be written as an implicit Runge–Kutta method, and it is numerically stable. The method was developed by John Crank and Phyllis Nicolson in the mid 20th century. For diffusion equations (and many other equations), it can be shown the Crank–Nicolson method is unconditionally stable. However, the approximate solutions can still contain (decaying) spurious oscillations if the ratio of time step \Delta t times the thermal diffusivity to the square of space step, \Delta x^2, is large (typically, larger than 1/2 per Von Neumann stability analysis). For this reason, whenever large time steps or high spatial resolution is necessary, the less accurate backward Euler method is often used, which is both stable and immune to oscillations. The method ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer-aided Engineering
Computer-aided engineering (CAE) is the broad usage of computer software to aid in engineering analysis tasks. It includes , , , durability and optimization. It is included with computer-aided design (CAD) and computer-aided manufacturing (CAM) in the collective abbreviation " CAx". Overview Computer-aided engineering primarily uses computer-aided design (CAD) software, which are sometimes called CAE tools. CAE tools are used, to analyze the robustness and performance of components and assemblies. CAE tools encompass simulation, validation, and optimization of products and manufacturing tools. CAE systems aim to be major providers of information to help support design teams in decision-making. Computer-aided engineering is used in various fields, like automotive, aviation, space, and shipbuilding industries. CAE systems can provide support to businesses. This is achieved by the use of reference architectures and their ability to place information views on the business proc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Explicit And Implicit Methods
Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes. ''Explicit methods'' calculate the state of a system at a later time from the state of the system at the current time, while ''implicit methods'' find a solution by solving an equation involving both the current state of the system and the later one. Mathematically, if Y(t) is the current system state and Y(t+\Delta t) is the state at the later time (\Delta t is a small time step), then, for an explicit method : Y(t+\Delta t) = F(Y(t))\, while for an implicit method one solves an equation : G\Big(Y(t), Y(t+\Delta t)\Big)=0 \qquad (1)\, to find Y(t+\Delta t). Computation Implicit methods require an extra computation (solving the above equation), and they can be much harder to implement. Implicit methods are used because many pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
