Assume a problem with an infinite number of unknowns. The finite element discretization procedures reduce the problem to one of a finite number of unknowns: dividing the solution region into discrete elements expressing the unknown field variable in terms of assumed approximating functions within each element Notes The discrete elements can be used to represent exceedingly complex geometric shapes, since these elements can be put together in various ways The approximating functions( or interpolation functions) are defined in terms of the values of the field variables at specified points called nodes Nodes usually lie on the element boundaries where adjacent elements are considered to be connected Notes Clearly, the nature of the solution and the degree of approximation depend not only on the size and number of the element used, but also on the interpolation functions selected In essence, a complex problem reduces to considering a series of greatly simplified problems FEM Steps To summarize in general terms how the finite element method works, the following steps are listed 1 Discretize the continuum. The step is to divide the continuum or solution region into elements 2. Select interpolation functions. The next step is to assign nodes to each element and then choose the type of nterpolation function to represent the variation of the field variable over the element 3. Find the element properties. Once the finite element model has been established, the matrix equations can be determined, thus expressing the properties of the individual elements Assemble the element properties to obtain the system equations. Combine the matrix equations expressing the behavior of the elements and form the matrix equations expressing the behavior of the entire solution region or system 5. Solve the system equations. The assembly process of the preceding step gives a set of simul taneous equations that must be solved to obtain alue of the field ariable 6. Make additional computation if desired. Sometimes it is desirable to use the solution of the system equations to calculate other important parameters Example 1(ANSYS)Example 2( Crash Analysis for a Car) 8. Summary A solid material under applied load will deform. This deformation may be under tension, compression shear, and torsion, depending on how the loads are applied. It is also possible that one or more of these loading In the realm of engineering design, it is important to know the behavior of a structure under applied loads. Unsafe designs are characterized by extensive deform-ations. In such cases, the engineer must redesign the component or structure 4 The theory of solid mechanics is used to predict the amount of deformation under applied loads and hence whether or not a design is safeAssume a problem with an infinite number of unknowns. The finite element discretization procedures reduce the problem to one of a finite number of unknowns: dividing the solution region into discrete elements； expressing the unknown field variable in terms of assumed approximating functions within each element. Notes The discrete elements can be used to represent exceedingly complex geometric shapes, since these elements can be put together in various ways. . The approximating functions ( or interpolation functions) are defined in terms of the values of the field variables at specified points called nodes. Nodes usually lie on the element boundaries where adjacent elements are considered to be connected. Notes Clearly, the nature of the solution and the degree of approximation depend not only on the size and number of the element used, but also on the interpolation functions selected. . .In essence, a complex problem reduces to considering a series of greatly simplified problems. FEM Steps To summarize in general terms how the finite element method works, the following steps are listed. 1 、 Discretize the continuum. The first step is to divide the continuum or solution region into elements. . 2、Select interpolation functions. The next step is to assign nodes to each element and then choose the type of interpolation function to represent the variation of the field variable over the element. 3. Find the element properties. Once the finite element model has been established, the matrix equations can be determined, thus expressing the properties of the individual elements. . 4. Assemble the element properties to obtain the system equations. Combine the matrix equations expressing the behavior of the elements and form the matrix equations expressing the behavior of the entire solution region or system. 5. Solve the system equations. The assembly process of the preceding step gives a set of simultaneous equations that must be solved to obtain the unknown nodal value of the field variable. . 6. Make additional computation if desired. Sometimes it is desirable to use the solution of the system equations to calculate other important parameters. Example 1 (ANSYS) Example 2（Crash Analysis for a Car） 8. Summary ❖ A solid material under applied load will deform. This deformation may be under tension, compression, shear, and torsion, depending on how the loads are applied. It is also possible that one or more of these loading cases may exist simultaneously. . ❖ In the realm of engineering design, it is important to know the behavior of a structure under applied loads. Unsafe designs are characterized by extensive deform-ations. In such cases, the engineer must redesign the component or structure. . ❖ The theory of solid mechanics is used to predict the amount of deformation under applied loads and hence whether or not a design is safe.