Finite Element Method (FEM) - Finite Element Analysis (FEA): Easy Explanation

an engineer designing a bridge will need to know how the proposed structure will behave under load the equation scrubbing the distribution of structural stresses are known but they can't be directly solved for a complicated shape such as a bridge however the equations can be solved for very so the finite element method takes it replace the single complicated shape with an approximately equivalent network of simple elements the overall pattern of elements is referred to as the finite element mesh and this pattern will be unique to each new problem the initial step is to design this mesh and for this we must first decide what kind of elements will use one-dimensional rods or two-dimensional triangles or quadrilaterals all three-dimensional blocks are a combination of different kinds of elements the accuracy of the calculation is going to depend on the number of elements we choose to have in the mesh the more elements we have the smaller each one will be and the more accurate the results unfortunately more elements also means more calculations to be done for reasons of economy therefore we look for the happy medium just enough elements to give adequate accuracy within a reasonable computing time in our example of the bridge we've chosen to use elements defined by eight points these points are referred to as nodes in general we regard each node has been capable of moving both horizontally and vertically the exception will be those points around the outside edge that we can regard as immovable fixed these boundary conditions must be included to complete the description of a physical problem so that the solution will be uniquely defined finally we must specify the elastic properties of the material what loading we wish to apply in this example we've chosen a large point load at the center of the bridge the mathematical treatment aims at deriving an equation describing the whole system we start from a basic relationship expressing the displacement of any node as a function of the nodes coordinates x and y in this example the elements have 8 nodes and for each of these we write an equation describing the displacement of the node as a function of its coordinates this gives us eight equations for the whole element which together form a matrix this matrix is now the starting point for a series of steps based on the fundamental laws of mechanics step one relates displacements to stresses from stresses we obtain strain energy and from this we derive potential energy and finally from minimum potential energy we obtain a pair of system equations for the complete element by analogy with the equation of a simple spring this new matrix is called the stiffness matrix for the element instead of a single displacement X the matrix operates on the vector X whose components are the displacements for the whole element we carry out this process for every element in the mesh so that we now have a stiffness matrix for each one the important step now is to combine all these individual matrices into a single large matrix representing the stiffness of the whole system now any two neighbouring elements will have nodes in common so values for common nodes will appear in both matrices the matrices can therefore be combined by a simple merging technique in practice the process of solving the overall system equation is done concurrently with the combining of the matrices the whole process is known as reduction we now use a standard procedure to eliminate part of the matrix the rows of the matrix represent a set of simultaneous equations we solve the first equation and substitute the solution in the remaining ones repeating until finally we're ready to add on the matrix for the next element eventually when the last matrix has been added we're left with the solution for a single node we can now use this result rather like a key working backwards through all the equations of the system until at last the displacement of every node is obtained from these results we can quickly calculate the corresponding stresses in this example we're showing the stresses as a contour diagram with the areas of greatest compression shown in red and greatest tension in dark blue the process we've been watching requires vast numbers of individual calculations to be done and for this reason can only be done on the computer furthermore the calculations usually involve such large matrices that a powerful computer with a large core store is needed a great advantage of the computer however is that now we've completed the calculation for a load at one point it's a very simple matter to repeat the whole thing with the load move to another point if we keep repeating the process with a sequence of different points we can eventually build up a continuous animated sequence showing the behavior of the bridge as the load moves across it this particular sequence was calculated using the Pethick finite elements computer program developed in the mechanical engineering department at Nottingham University and was then drawn onto film using the antics graphic animation program at the Atlas computer laboratory many other users of the Atlas Laboratory have also been working on applications of the finite element method the civil engineering department at Southampton University is developing a computer program to study the behavior of fluids in motion when fully developed this program could find important practical applications in the design of offshore structures such as North Sea oil rigs in this example we're simulating a channel containing a rectangular block obstruction the finite element mesh is based on triangular elements we introduce a steady flow of liquid into the channel the speed of flow the viscosity of the liquid and the dimensions of the channel are all specified the upper diagram shows the path of the fluid the distribution of free stream lines the lower diagram shows what are called stationary stream lines it contains the same information as the upper diagram except that the uniform velocity of unobstructed steady flow has been subtracted in such a way as to exaggerate and clarify the small variations we wish to study at this point we've made a change in the conditions equivalent to speeding up the flow the stream lines change until again they form a steady state increasing the speed of flow still further however and a steady state is no longer possible vortices begin to form and eventually a dynamic equilibrium state is reached finally one more example also from Southampton this program is designed to study the dispersion of pollutants deposited in tidal waters the finite element mesh here corresponds to the eastern Solent extending to Portsmouth and the thickness of the elements corresponds to the depth of the water we also include the rise and fall of the tide and insert a specified outfall of pollutants the computer program then shows us clearly whether the pollutants will be carried straight out to sea or whether tidal effects will cause dangerous concentrations to build up near the beaches once again from this example we could now continue the same program may be used for any other area of water and any specified discharge of pollutants we can investigate varying the rate of discharge and timing it to fit in with the movement of the tide we can take into account the biological decay of the pollutant and we can also include the background level of pollution from the rivers themselves a major advantage of the finite element method is the ability to work with arbitrary shapes this overcomes many of the limitations of older numerical techniques and gives the method of flexibility comparable to the use of physical models but without their limitations for example different materials can be investigated easily by specifying different material properties as programmed data properties that are difficult to vary in physical models we can simulate life-size structures on the computer whereas with the use of small scale models we're left with the scaling up problems engineering research today is becoming increasingly sophisticated it often only by using the largest and fastest computers that we can help to solve the complicated systems of equations representing a complex physical situation with modern powerful computers it's also possible to conduct a comprehensive parameter search using a finite element program for large numbers of different cases in order to optimize the system the finite element method can be applied to the widest possible variety of engineer undoubtedly it can be said to lead the field in modern engineering computing methods [Music]

Channel: Educational Video Library

Views: 190,628

Keywords: Finite, Element, Method, FEM, Finite Element Analysis, FEA, numerical, solving, engineering, Mathematical physics

Id: aLJMDn_2-d8

Length: 10min 28sec (628 seconds)

Published: Mon Mar 20 2017