Finite Element Methods
Finite Element Methods (FEM) are numerical techniques used to approximate solutions to complex physical problems described by differential equations, such as heat conduction, fluid flow, and structural analysis. By dividing the problem domain into smaller, manageable sections known as a mesh, FEM simplifies the original equations into a system that can be solved computationally. Each section of the mesh allows for the application of piecewise polynomial functions to model the physical behavior more accurately. This method is especially useful in engineering fields, where it helps in designing structures like buildings and aircraft, ensuring they can withstand various forces and environmental effects.
The origins of FEM can be traced back to the work of mathematicians such as Richard Courant and later developed by others in the mid-20th century. The method involves assembling stiffness matrices and load vectors to create a system of equations that represent the physical scenario. As computational capabilities have advanced, FEM has become integral in simulating and analyzing dynamic processes over time, making it a vital tool in engineering design and safety assessment. Overall, FEM provides a robust framework for solving real-world engineering challenges by offering precise approximations of complex physical phenomena.
Subject Terms
Finite Element Methods
Type of physical science: Computation
Field of study: Numerical methods
The equations that model physical processes, such as a vibrating string or heat conduction in a rod, can be difficult to solve. Finite element methods approximate the original model with a simpler system of equations which can be solved on a computer.
Overview
Physical processes such as liquid flow in a channel, heat conduction in a rod, or the vibration of an elastic membrane are modeled mathematically by differential equations. Since differential equations can be difficult to solve, techniques to approximate their solutions have been devised. The finite element method is one popular approximation technique.
To illustrate the finite element approximation, consider an elastic string which is fastened at two points, A and B, and which is pushed from below. If h(x) denotes the height of the string n at a point x inches from A; then h(x) satisfies an equation of the following form: -T(d²h(x)/dx²)=f(x) where the constant T measures the elastic properties of the string and f(x) is the external force being applied from below. The expression d²h(x)/dx² stands for the second derivative of h with respect to x and is related to the curvature of the string. For a straight string, the second derivative is zero; for a wiggly string, the second derivative is large in magnitude. Physically, this equation expresses the balance of force; that is, the force related to the tension in the string is equal to the external force. Any equation that contains derivatives is called a differential equation.
One of the simplest finite elements is the piecewise linear shape function. This function is piecewise linear since it is composed of several different line segments patched together. Although the shape of the string in figure 1 is quite different from the piecewise linear function in figure 2, the piecewise linear functions can be added together in order to obtain a much better approximation of the smooth shape of figure 1. The approximation process works in the following way. The interval between A and B, called the problem domain, is divided into several pieces, such as into four pieces. These pieces form what is called a mesh, and the boundary points x1, x2, and x3 between two adjacent mesh intervals are the nodes that are associated with the piecewise linear finite element space.
Consider the three piecewise linear finite elements depicted in figure 3. Although each of these finite elements by itself does not resemble the shape in figure 1, the sum of the three graphs, obtained by adding together the vertical heights at each point between A and B, is an approximate shape. If a finer mesh is employed with more mesh intervals, then an even better approximation to the smooth curve of figure 1 can be obtained.
In constructing the finite element approximation of figure 4, it was assumed that the curve h(x) was known and a piecewise linear approximation was exhibited. In practice, however, the curve h(x) must be obtained by solving the differential equation. In approximation schemes such as the finite element method, the approximation solution, such as the one shown in figure 4, is generated by solving equations that, in some sense, model the original differential equation. Let h1, h2, and h3 denote the height of the string at the mesh points x1, x2, and x3. The finite element equations that are associated with piecewise linear finite elements are the following: T(2h1 - h2) = Δx²f1, T(h1 + 2h2 - h3) = Δx²f2, + T(h2 + 2h3) = Δx²f3, where Δx stands for the width of the mesh intervals. Notice that there are three equations in the three unknowns h1, h2, and h3. The terms multiplied by T correspond to the stiffness term in the original differential equation, while the forces at the mesh points are denoted f1, f2, and f3, respectively. The coefficients of the stiffness terms in the finite element equation are often arranged in the following table:
Multiple line equation(s) cannot be represented in ASCII text; please see PDF of this article if available.
There is a 2 in row 1 and column 1 because h1 in the first finite element equation is multiplied by 2. There is a -1 in row 1 and column 2 because h2 in the first finite element equation is multiplied by -1. There is a 0 in row 1 and column 3 because h3 does not appear in the first equation. A table of numbers such as this is called a matrix, and this particular matrix is a finite element stiffness matrix because it is connected with the stiffness term in the original differential equation. The load (force) at the mesh points can be arranged in a column called the load vector, shown as the following:
Multiple line equation(s) cannot be represented in ASCII text; please see PDF of this article if available.
Systems of equations like the three equations and three unknowns appearing above are solved routinely by computers. After computing numerical values for the heights h1, h2, and h3 at the nodes x1, x2, and x3, one can construct the piecewise linear finite element approximation shown in figure 4 by connecting the computed points with line segments.
In this particular example, the mass of the string does not appear in the modeling equation. In more complicated physical systems, where the mass must be accounted for in the mathematical model, an additional term appears in the differential equation corresponding to the mass. In finite element equations, the matrix associated with this additional term is called the mass matrix.
Applications
The finite element method is used to approximate the solution to mathematical models that arise in almost every branch of science and engineering. In each application, the physical process occupies some region of space referred to as the problem domain. The domain is partitioned into a mesh. For a one-dimensional string, the mesh consists of line segments. For a two-dimensional application, such as a model for the surface of a drum, the mesh is often constructed from small triangles or rectangles. In a three-dimensional application, such as a model for a building, the mesh often consists of either small tetrahedrons or boxes. On different components of the mesh, the solution to the physical process is approximated by different polynomials. These collections of polynomials are patched together to obtain a piecewise polynomial approximation for the solution over the entire domain. In order to construct these polynomials and to form the composite piecewise polynomial, a system of equations is solved on a computer, yielding the value of the polynomials at the nodes in the mesh.
Finite element methods play crucial roles in the modeling of elastic structures. An example of an elastic structure is a tall building. As the wind blows during a storm or as the earth vibrates during an earthquake, the building sways. An engineer designs the building to be strong enough to withstand wind of a specified velocity and earth movements of a given magnitude. In order to determine whether the design is strong enough to withstand the prescribed wind and earth movement, the engineer formulates a mathematical model for the building, for the wind, and for an earthquake. By studying the finite element solution to the model, the engineer can determine whether the building design is sturdy enough.
In the aerospace industry, the finite element method is used to model the flow of air around the wings and the stresses and strains throughout the aircraft. In designing aircraft, it is important to minimize the drag forces on the plane: As the drag increases, the plane slows down and more fuel is expended during a flight. By examining the finite element solutions corresponding to different designs, an engineer can determine the aircraft shapes that are most efficient. Similarly, in the automobile industry, engineers use the finite element to study the drag on an automobile as it travels down the road. Again, by studying finite element solutions, an engineer can determine which automobile shapes are the most aerodynamic. A better aerodynamic shape leads to better fuel economy.
In some applications, it is important to determine how the physical process evolves in time. For example, suppose that a hot steel bar were placed in a tub of water. With time, the temperature profile in the bar would change. The differential equation that describes the conduction of heat is called the heat equation. The finite element method can be used to determine the temperature profile at any given time. Thus, for each instant of time, there is a different finite element solution corresponding to the temperature distribution throughout the rod.
Context
In the work of Baron Rayleigh (John William Strutt) in 1870 and Walther Ritz in 1908, the solutions of differential equations were approximated by simple functions that extended over the entire problem domain. In honor of their contributions, methods based on approximations over the entire domain are called Rayleigh-Ritz methods. In finite element methods, the problem domain is partitioned into subdomains, and simple functions are used to approximate the solution over the subdomains. Except for this partitioning of the domain into a mesh, the Rayleigh-Ritz method is quite similar to finite element methods.
The term "finite element method" is attributed to Gerald Wayne Clough, who used it in a paper in 1970. The mathematical theory of a finite element method dates back to the work of Richard Courant in 1943. In the work of Alexander Paul Hrennikoff in 1941 and Douglas McHenry in 1943, the finite element method was first used to analyze stress and strain in elastic structures.
Principal terms:
DIFFERENTIAL EQUATIONS: equations that contain the derivatives of functions; often arise in modeling physical processes such as the vibration of elastic structures, the conduction of heat, and fluid flow
LOAD VECTORS: the external forces
MASS MATRIX: the term in finite element equations that is related to bending properties
MESH: a subdivision of the problem into smaller subdomains
NODES: critical points associated with the shape function that uniquely specify what the shape function looks like
PIECEWISE POLYNOMIAL: a polynomial, such as a line or quadratic, which is patched together in order to obtain a shape function
PROBLEM DOMAIN: the region in space that is associated with a given physical process
SHAPE FUNCTION: an object with a simple shape which is used to approximate the physical process with small subdomains
STIFFNESS MATRIX: the term in finite element equations that is related to bending properties
Bibliography
Becker, Eric B., Graham F. Carey, and J. Tinsley Oden. FINITE ELEMENTS. Vols. 1-6. Englewood Cliffs, N.J.: Prentice-Hall, 1981-1986. This series of books provides a very comprehensive treatment of finite element methods, focusing on computational aspects, mathematical aspects, applications in solid mechanics, and applications in fluid mechanics.
Ciarlet, P. G. THE FINITE ELEMENT METHOD FOR ELLIPTIC PROBLEMS. Amsterdam: North-Holland, 1978. This book provides a thorough mathematical analysis of the finite element method. The presentation is oriented to mathematics.
Hall, Charles A., and T. A. Porsching. NUMERICAL ANALYSIS OF PARTIAL DIFFERENTIAL EQUATIONS. Englewood Cliffs, N.J.: Prentice-Hall, 1990. Provides a broad view of numerical methods for differential equations. Finite element methods are among the many methods studied. Both authors are mathematicians, and the treatment tends to be mathematically oriented. There are some sections, however, that provide descriptions from the perspective of the engineer or practitioner.
Strang, G., and G. J. Fix. AN ANALYSIS OF THE FINITE ELEMENT METHOD. Englewood Cliffs, N.J.: Prentice-Hall, 1973. This book is written by mathematicians who had an engineering audience in mind. Provides excellent insight into the finite element method without immersing the reader in many of the technical mathematical issues.
Zienkiewicz, O. C. THE FINITE ELEMENT METHOD. Maidenhead, Berkshire, England: McGraw-Hill, 1977. In this classic reference, the finite element method is presented in an engineering context, with particular emphasis on finite element methods for elastic structures.
Displacement of an elastic string by an external force
Piecewise linear shape function
Mesh, node, and piecewise linear shape functions
Piecewise linear approximation of the elastic string
Numerical Solutions of Differential Equations