A numerical method that divides a body into small elements and solves simultaneous equations with nodal displacements or similar as unknowns to find approximate distributions of stress or temperature.
Japanese name: 有限要素法
Finite element method appears in 2 practice questions across 1 topic, including Computational materials science.
The finite element method divides a body of complex shape into a collection of simple elements such as triangles, tetrahedra and hexahedra, and solves the whole by assembling equations approximated element by element. The original differential equations are recast in the form of the principle of minimum energy or virtual work (the weak form), so boundary shapes and differences in material are handled naturally. It grew out of aircraft structural analysis in the 1950s, and the name finite element method is generally traced to its use by Clough in 1960. Today it is used not only for stress but also for heat conduction, electromagnetic fields, fluids and coupled problems in which these interact.
Linear elements, which interpolate the displacement within an element linearly from nodal values, are cheap, but in regions under bending they cannot represent the strain correctly and can come out stiffer than reality (shear locking). Materials whose volume hardly changes, such as rubber or plastically deforming metal, likewise come out too stiff through volumetric locking. Quadratic elements and reduced-integration elements with fewer integration points ease this, but reduced integration can produce zero-energy hourglass-shaped deformation (hourglass modes), which must also be suppressed. Because results depend on the element used, it is important to choose elements suited to the problem.
The finite element method only solves the equations under the conditions given; it does not guarantee that those conditions match reality. Large errors often come not so much from the mesh as from the boundary conditions (where and how the body is fixed, and where and how loads are applied) and from the material data. For example, applying a concentrated load at a single point, or using a shape with sharp corners as it is, makes the stress at that point grow as the mesh is refined, so it does not converge to a meaningful value. Results are checked against theoretical solutions for simple shapes and against measurements such as strain gauges.
In materials work it is used heavily for nonlinear problems such as plasticity, creep and thermal stress, not just elasticity. Typical applications are process simulations: material flow and springback in forging and press forming, temperatures and residual stresses in welding and heat treatment, and the solidification of castings. In the crystal plasticity finite element method, each grain is represented by elements and deformation on each slip system is calculated, allowing the development of texture and the uneven distribution of strain among grains to be studied. In composites, it is also used for homogenization, calculating a representative volume element that resolves the fibers and matrix to find the average properties of the whole.
Answer: C. It is an approximate method, so elements are refined where stress concentrates and it is checked that the result hardly changes with element size (convergence)
In the finite element method, the structure is divided into small elements such as triangles and quadrilaterals, and the displacement within each element is interpolated from the nodal displacements with simple functions. The global stiffness equation [K]{u} = {F} is solved for the nodal displacements, from which strains and stresses are calculated. Because the interpolation is approximate, the basic practice is to refine the elements near holes and notches where the stress changes sharply, and to check that the values stop changing as the mesh is refined. Coarser elements normally give larger errors. On the other hand, in linear elasticity the stress at the tip of a sharp crack is theoretically infinite, so the value keeps growing as the mesh is refined and never converges. Cracks are therefore evaluated not by stress but by the stress intensity factor or the J-integral, with singular elements or similar used at the tip.
The finite difference method replaces derivatives with differences at the points of a regular grid; it is easy to set up and fast for regions of simple shape. The finite element method allows the shape and size of elements to be chosen freely, so it suits parts of complex shape and problems where the fineness should vary from place to place.
An explicit method finds the next state directly from the current one, so each step is light, but the time step must be very small for stability. An implicit method solves simultaneous equations, so each step is heavy, but large steps can be used. Explicit methods are often used for impact and high-speed forming, and implicit methods for static problems.
Elastic calculations need Young's modulus and Poisson's ratio; thermal calculations need thermal conductivity, specific heat and density; thermal stress also needs the coefficient of linear expansion. Plasticity needs the stress-strain curve, and creep the constants of the rate equation, entered including their variation with temperature and strain rate.
In the finite element method, the structure is divided into small elements such as triangles and quadrilaterals, and the displacement within each element is interpolated from the nodal displacements with simple functions.
Which is the correct way of thinking when using the finite …The phase-field method treats microstructural change from nanometers to micrometers, and the finite element method treats stress and temperature at the scale of components as a continuum.
Which pairing of a materials computation method with the ra…Stress intensity factor Density functional theory Molecular dynamics Interatomic potential Phase-field method CALPHAD method Materials informatics Monte Carlo method