Finite element method

We can study big things by breaking them down.

We cut one big thing into many small parts. These small parts are called elements.

A computer uses these parts to solve hard puzzles. It can help us see how heat moves. It can show how water flows. This helps us test things without building them. It can even help us predict the weather. Using these small pieces makes big jobs easier. It is a smart way to learn about our world.

Sometimes, big problems are too hard to solve all at once. Imagine trying to study how a whole car reacts to a crash.

Instead of looking at the whole car, we use a special way called the finite element method. This method, or FEM, breaks a large object into many small, simple parts. We call these small parts finite elements.

To do this, we build a mesh. A mesh is a group of points that covers the object. Each small part has its own simple math equations. Then, a computer puts all those small equations back together. This creates one big system of equations for the whole object.

Using FEM helps us study many things. It can show how heat moves or how fluids flow. It can even help predict the weather, like tracking big storms.

This method is very useful for engineers. It lets them test new ideas on a computer. They do not have to build every single prototype to see if it works. This saves a lot of time and money.

Sometimes, big problems are too hard to solve all at once. Imagine trying to study how a whole car reacts to a crash.

Instead of looking at the whole car, we use a special way called the finite element method. This method, or FEM, breaks a large object into many small, simple parts. We call these small parts finite elements.

By dividing a large system into these smaller pieces, we can study complex shapes more easily. This makes it much simpler to represent tricky shapes or different materials. It also helps us see what is happening in one tiny spot.

To make this work, we must build a mesh. A mesh is a collection of points that covers the object.

Each small element has its own simple math equations. These equations are called element equations. They help approximate the much harder equations of the whole object. A computer then takes all these small equations and puts them back together. This creates one giant system of equations for the entire problem. This big system can then be solved to find a numerical answer.

People have been working on this for a long time. The method grew from a need to solve hard problems in engineering. In the early 1940s, Alexander Hrennikoff and Richard Courant did important early work. Courant used triangular shapes to study how a cylinder twists. Other pioneers include Ioannis Argyris and Leonard Oganesyan. In China, Feng Kang also found this method in the late 1950s. He used it to study how dams are built.

Many groups helped make this method famous in the 1960s and 1970s. Scientists at the University of Stuttgart and UC Berkeley worked on it. Even NASA helped by sponsoring an early version called NASTRAN. In 1969, a group in Norway made a program called Sesam for ships.

Later, in 1973, Gilbert Strang and George Fix gave the method a strong math basis. Today, we use high-speed supercomputers to solve the biggest and most complex problems.

This method is very useful for many different jobs. Engineers use it to test things like oil pipelines or heat transfer. It can even help predict the weather. It is great for tracking big storms like tropical cyclones.

Using a computer simulation is much better than building many expensive prototypes. For example, we can test a car crash on a screen. We can make the math more exact for the front of the car. This helps us save a lot of time and money.

The finite element method (FEM) is a powerful numerical technique used to solve complex mathematical problems. It is specifically designed to solve partial differential equations (PDEs) that arise in engineering and physics. These equations describe how things change across space and time. Engineers use FEM to study structural analysis, heat transfer, fluid flow, and electromagnetism. Because these problems are often too difficult to solve with exact formulas, computers perform the necessary calculations. High-speed supercomputers are often required to handle the largest and most complex simulations.

To solve a large problem, FEM uses a process called discretization. This involves subdividing a large, complex system into much smaller, simpler parts called finite elements. This process creates a numerical domain known as a mesh, which consists of a finite number of points. Dividing a system into subdomains offers several technical advantages. It allows for an accurate representation of complex geometries and the inclusion of dissimilar material properties. It also makes it easier to represent the total solution and capture local effects within the system.

The mechanism of FEM follows a specific sequence of mathematical steps. First, the original problem is divided into subdomains, each represented by a set of element equations. These element equations are simpler versions that locally approximate the original, complex partial differential equations. In many cases, FEM is viewed as a special case of the Galerkin method. This involves constructing an integral of the inner product of a residual and weight functions. The goal is to minimize the approximation error by fitting trial functions into the PDE. The process eliminates spatial derivatives, resulting in algebraic equations for steady-state problems or ordinary differential equations for transient problems.

Once the element equations are created, they must be reassembled. This second step involves generating a global system of equations by transforming coordinates from local nodes to global nodes. This spatial transformation includes making orientation adjustments relative to a reference coordinate system. The simple equations from each element are then assembled into one massive system that models the entire problem. Computers use known solution techniques to solve this global system and obtain a numerical answer.

History shows that FEM grew from the need to solve structural analysis problems in civil and aeronautical engineering. Its roots can be traced to the early 1940s with the work of Alexander Hrennikoff and Richard Courant. Hrennikoff used a lattice analogy to discretize domains, while Courant divided domains into triangular sub-regions. Courant's work helped solve equations related to the torsion of a cylinder. Other pioneers included Ioannis Argyris and Leonard Oganesyan. In the late 1950s, Feng Kang independently rediscovered the method in China while calculating dam constructions.

The method gained significant momentum during the 1960s and 1970s. Various research groups at institutions like the University of Stuttgart, UC Berkeley, and Swansea University contributed to its development. During this era, NASA sponsored the original version of the NASTRAN program. In 1969, the Norwegian society Det Norske Veritas developed Sesam for ship analysis. A rigorous mathematical foundation for the method was finally established in 1973 by Gilbert Strang and George Fix.

Today, the practical application of FEM is known as finite element analysis (FEA). FEA is a vital computational tool that allows engineers to test physical systems without building expensive prototypes. For example, in a frontal car crash simulation, engineers can increase prediction accuracy at the front of the car while reducing it at the rear to save costs.

This method is also used in numerical weather prediction. It is especially useful for modeling highly nonlinear phenomena, such as tropical cyclones in the atmosphere or eddies in the ocean.

By using FEM, scientists can model everything from oil pipelines to the complex mechanics of a human knee joint.