A discrete element method (DEM), also called a distinct element method, is any of a family of numerical methods for computing the motion and effect of a large number of small particles. Though DEM is very closely related to molecular dynamics, the method is generally distinguished by its inclusion of rotational degrees-of-freedom as well as stateful contact and often complicated geometries. With advances in computing power and numerical algorithms for nearest neighbor sorting, it has become possible to numerically simulate millions of particles on a single processor. Today DEM is becoming widely accepted as an effective method of addressing engineering problems in granular and discontinuous materials, especially in granular flows, powder mechanics, and rock mechanics. Recently, the method was expanded into the Extended Discrete Element Method taking thermodynamics and coupling to CFD and FEM into account.
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 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.
Numerical partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations. including fluid mechanics, acoustics, electromagnetics, fracture mechanics, and contact mechanics.
Mesh generation is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells. Usually these cells form a simplicial complex, yet meshes are distinguished by having a geometric embedding. Usually the cells partition the geometric input domain. Mesh cells are used as discrete local approximations of the larger domain. Meshes are created by computer algorithms, often with human guidance through a GUI, depending on the complexity of the domain and the type of mesh desired. The goal is to create a mesh that accurately captures the input domain geometry, with high-quality (well-shaped) cells, and without so many cells as to make subsequent calculations intractable. The mesh should also be fine in areas that are important for the subsequent calculations.
Computational electromagnetics, computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment.
Flow visualization or flow visualisation in fluid dynamics is used to make the flow patterns visible, in order to get qualitative or quantitative information on them.
Discontinuous deformation analysis (DDA) is a type of discrete element method (DEM) originally proposed by Shi in 1988. DDA is somewhat similar to the finite element method for solving stress-displacement problems, but accounts for the interaction of independent particles (blocks) along discontinuities in fractured and jointed rock masses. DDA is typically formulated as a work-energy method, and can be derived using the principle of minimum potential energy or by using Hamilton's principle. Once the equations of motion are discretized, a step-wise linear time marching scheme in the Newmark family is used for the solution of the equations of motion. The relation between adjacent blocks is governed by equations of contact interpenetration and accounts for friction. DDA adopts a stepwise approach to solve for the large displacements that accompany discontinuous movements between blocks. The blocks are said to be "simply deformable". Since the method accounts for the inertial forces of the blocks' mass, it can be used to solve the full dynamic problem of block motion.
In the numerical solution of partial differential equations, a topic in mathematics, the spectral element method (SEM) is a formulation of the finite element method (FEM) that uses high degree piecewise polynomials as basis functions. The spectral element method was introduced in a 1984 paper by A. T. Patera.
In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs and finite element meshes. DEC methods have proved to be very powerful in improving and analyzing finite element methods: for instance, DEC-based methods allow the use of highly non-uniform meshes to obtain accurate results. Non-uniform meshes are advantageous because they allow the use of large elements where the process to be simulated is relatively simple, as opposed to a fine resolution where the process may be complicated, while using less computational power than if a uniformly fine mesh were used.
The finite element method (FEM), is a numerical method for solving problems of engineering and mathematical physics. Typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The analytical solution of these problems generally require the solution to boundary value problems for partial differential equations. The finite element method formulation of the problem results in a system of algebraic equations. The method approximates the unknown function over the domain. To solve the problem, it subdivides a large system into smaller, simpler parts that are called finite elements. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function.
hp-FEM is a general version of the finite element method (FEM), a numerical method for solving partial differential equations based on piecewise-polynomial approximations that employs elements of variable size (h) and polynomial degree (p). The origins of hp-FEM date back to the pioneering work of Ivo Babuska et al. 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-refinements. The exponential convergence makes the method a very attractive choice compared to most other finite element methods which only converge with an algebraic rate. The exponential convergence of the hp-FEM was not only predicted theoretically but also observed by numerous independent researchers.
In applied mathematics, the name finite pointset method is a general approach for the numerical solution of problems in continuum mechanics, such as the simulation of fluid flows. In this approach the medium is represented by a finite set of points, each endowed with the relevant local properties of the medium such as density, velocity, pressure, and temperature.
Fluid motion is governed by the Navier–Stokes equations, a set of coupled and nonlinear partial differential equations derived from the basic laws of conservation of mass, momentum and energy. The unknowns are usually the flow velocity, the pressure and density and temperature. The analytical solution of this equation is impossible hence scientists resort to laboratory experiments in such situations. The answers delivered are, however, usually qualitatively different since dynamical and geometric similitude are difficult to enforce simultaneously between the lab experiment and the prototype. Furthermore, the design and construction of these experiments can be difficult, particularly for stratified rotating flows. Computational fluid dynamics (CFD) is an additional tool in the arsenal of scientists. In its early days CFD was often controversial, as it involved additional approximation to the governing equations and raised additional (legitimate) issues. Nowadays CFD is an established discipline alongside theoretical and experimental methods. This position is in large part due to the exponential growth of computer power which has allowed us to tackle ever larger and more complex problems.
The finite point method (FPM) is a meshfree method for solving partial differential equations (PDEs) on scattered distributions of points. The FPM was proposed in the mid-nineties in, and with the purpose to facilitate the solution of problems involving complex geometries, free surfaces, moving boundaries and adaptive refinement. Since then, the FPM has evolved considerably, showing satisfactory accuracy and capabilities to deal with different fluid and solid mechanics problems.
In geology, numerical modeling is a widely applied technique to tackle complex geological problems by computational simulation of geological scenarios.
