• “[15] Generalized finite element method[edit] The generalized finite element method (GFEM) uses local spaces consisting of functions, not necessarily polynomials, that reflect
  the available information on the unknown solution and thus ensure good local approximation.

• A discretization strategy is understood to mean a clearly defined set of procedures that cover (a) the creation of finite element meshes, (b) the definition of basis function
  on reference elements (also called shape functions), and (c) the mapping of reference elements onto the elements of the mesh.

• The global system of equations has known solution techniques and can be calculated from the initial values of the original problem to obtain a numerical answer.

• Although the approaches used by these pioneers are different, they share one essential characteristic: mesh discretization of a continuous domain into a set of discrete sub-domains,
  usually called elements.

• Extended finite element methods enrich the approximation space to naturally reproduce the challenging feature associated with the problem of interest: the discontinuity, singularity,
  boundary layer, etc.

• So we now have to solve a linear system in the unknown where most of the entries of the matrix , which we need to invert, are zero.

• The finite element method formulation of a boundary value problem finally results in a system of algebraic equations.

• Typical work out of the method involves: 1. dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations
  to the original problem 2. systematically recombining all sets of element equations into a global system of equations for the final calculation.

• In applying FEA, the complex problem is usually a physical system with the underlying physics such as the Euler–Bernoulli beam equation, the heat equation, or the Navier-Stokes
  equations expressed in either PDE or integral equations, while the divided small elements of the complex problem represent different areas in the physical system.

• As we refine the triangulation, the space of piecewise linear functions must also change with .

• A piecewise linear function in two dimensions For problem P2[edit] We need to be a set of functions of .

• Basic concepts The subdivision of a whole domain into simpler parts has several advantages:[2] • Accurate representation of complex geometry • Inclusion of dissimilar material
  properties • Easy representation of the total solution • Capture of local effects.

• For instance, for a fourth-order problem such as , one may use piecewise quadratic basis functions that are .

• (b) The sparse matrix L of the discretized linear system(c) The computed solution, The primary advantage of this choice of basis is that the inner products and will be zero
  for almost all .

• In contrast, ordinary differential equation sets that occur in the transient problems are solved by numerical integration using standard techniques such as Euler’s method
  or the Runge-Kutta method.

• In the one-dimensional case, for each control point we will choose the piecewise linear function in whose value is at and zero at every , i.e., for ; this basis is a shifted
  and scaled tent function.

• However, this method of solving the boundary value problem (BVP) works only when there is one spatial dimension.

• FEA may be used for analyzing problems over complicated domains (like cars and oil pipelines) when the domain changes (as during a solid-state reaction with a moving boundary),
  when the desired precision varies over the entire domain, or when the solution lacks smoothness.

• For vector partial differential equations, the basis functions may take values in .

• When the errors of approximation are larger than what is considered acceptable, then the discretization has to be changed either by an automated adaptive process or by the
  action of the analyst.

• The weak form of P2[edit] If we integrate by parts using a form of Green’s identities, we see that if solves P2, then we may define for any by where denotes the gradient and
  denotes the dot product in the two-dimensional plane.

• [3] History While it is difficult to quote the date of the invention of the finite element method, the method originated from the need to solve complex elasticity and structural
  analysis problems in civil and aeronautical engineering.

• A clear, detailed, and practical presentation of this approach can be found in The Finite Element Method for Engineers.

• CutFEM[edit] The Cut Finite Element Approach was developed in 2014.

• [14] The approach is “to make the discretization as independent as possible of the geometric description and minimize the complexity of mesh generation, while retaining the
  accuracy and robustness of a standard finite element method.

• More advanced implementations (adaptive finite element methods) utilize a method to assess the quality of the results (based on error estimation theory) and modify the mesh
  during the solution aiming to achieve an approximate solution within some bounds from the exact solution of the continuum problem.

• The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems).

• One hopes that as the underlying triangular mesh becomes finer and finer, the solution of the discrete problem (3) will, in some sense, converge to the solution of the original
  boundary value problem P2.

• P1 is a one-dimensional problem where is given, is an unknown function of , and is the second derivative of with respect to .

• The process, in mathematical language, is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero.

• 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.

• XFEM[edit] Main article: Extended finite element method The extended finite element method (XFEM) is a numerical technique based on the generalized finite element method (GFEM)
  and the partition of unity method (PUM).

• However, the derivative exists at every other value of , and one can use this derivative for integration by parts.

• The function is the unique function of whose value is at and zero at every .

• After this second step, we have concrete formulae for a large but finite-dimensional linear problem whose solution will approximately solve the original BVP.

• Illustrative problems P1 and P2[edit] The following two problems demonstrate the finite element method.

• We define a new operator or map by using integration by parts on the right-hand-side of (1): where we have used the assumption that .

• To solve a problem, the FEM subdivides a large system into smaller, simpler parts called finite elements.

• [1] The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem.

• To explain the approximation in this process, the finite element method is commonly introduced as a special case of Galerkin method.

• Hence the convergence properties of the GDM, which are established for a series of problems (linear and nonlinear elliptic problems, linear, nonlinear, and degenerate parabolic
  problems), hold as well for these particular FEMs.

• General form of the finite element method[edit] In general, the finite element method is characterized by the following process.

• If this condition is not satisfied, we obtain a nonconforming element method, an example of which is the space of piecewise linear functions over the mesh, which are continuous
  at each edge midpoint.

• Under specific hypotheses (for instance, if the domain is convex), a piecewise polynomial of order method will have an error of order .

• Main article: Applied element method A-FEM[edit] Yang and Lui introduced the Augmented-Finite Element Method, whose goal was to model the weak and strong discontinuities without
  needing extra DoFs, as PuM stated.

• [24] Finite Element Model of a human knee joint[25] This powerful design tool has significantly improved both the standard of engineering designs and the design process methodology
  in many industrial applications.

• Comparison to the finite difference method The finite difference method (FDM) is an alternative way of approximating solutions of PDEs.

• The name virtual derives from the fact that knowledge of the local shape function basis is not required and is, in fact, never explicitly calculated.

• • There are reasons to consider the mathematical foundation of the finite element approximation more sound, for instance, because the quality of the approximation between
  grid points is poor in FDM.

• Various numerical solution algorithms can be classified into two broad categories; direct and iterative solvers.

• [11] The method has since been generalized for the numerical modeling of physical systems in a wide variety of engineering disciplines, e.g., electromagnetism, heat transfer,
  and fluid dynamics.

• [12][13] Technical discussion The structure of finite element methods[edit] A finite element method is characterized by a variational formulation, a discretization strategy,
  one or more solution algorithms, and post-processing procedures.

• It includes the use of mesh generation techniques for dividing a complex problem into small elements, as well as the use of software coded with a FEM algorithm.

• For higher-order partial differential equations, one must use smoother basis functions.

• • The quality of a FEM approximation is often higher than in the corresponding FDM approach, but this is highly problem-dependent, and several examples to the contrary can
  be provided.

• Hrennikoff’s work discretizes the domain by using a lattice analogy, while Courant’s approach divides the domain into finite triangular subregions to solve second order elliptic
  partial differential equations that arise from the problem of torsion of a cylinder.

• Choosing a basis[edit] Interpolation of a Bessel function 16 scaled and shifted triangular basis functions (colors) used to reconstruct a zeroeth order Bessel function J0
  (black) The linear combination of basis functions (yellow) reproduces J0 (black) to any desired accuracy.

• Generally, the higher the number of elements in a mesh, the more accurate the solution of the discretized problem.

• Post-processing procedures are designed to extract the data of interest from a finite element solution.

• The practical application of FEM is known as finite element analysis (FEA).

• It is increasingly being adopted by other commercial finite element software, with a few plugins and actual core implementations available.

• In step (2) above, a global system of equations is generated from the element equations by transforming coordinates from the subdomains’ local nodes to the domain’s global
  nodes.

• In contrast, computational fluid dynamics (CFD) tend to use FDM or other methods like finite volume method (FVM).

• If instead of making h smaller, one increases the degree of the polynomials used in the basis function, one has a p-method.

• High-order methods with large uniform p are called spectral finite element methods (SFEM).

• Mesh adaptivity may utilize various techniques; the most popular are: • moving nodes (r-adaptivity) • refining (and unrefined) elements (h-adaptivity) • changing order of
  base functions (p-adaptivity) • combinations of the above (hp-adaptivity).

 

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Photo credit: https://www.flickr.com/photos/aidanmorgan/3556123230/’]