# Automatic Differentiation: Techniques and Applications by Louis B. Rall (eds.)

By Louis B. Rall (eds.)

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In some MFree methods, we may also use the element mesh for background integration. 5 15 Shape Function Creation In FEM, shape functions are created based on elements, and therefore, the computation of shape functions has been straightforward. In the early years of the development of FEM, much of the work involved the formulation of all different types of elements. All the shape functions of finite elements satisfy the Kronecker delta function property. In MFree methods, however, the construction of shape functions has been and still is the central issue.

38) This is actually the original Hooke’s law. The Young’s modulus Ε can be obtained using a simple tensile test. 8. The difference is that the forces applied on a beam are transversal, that is, the direction of the force is perpendicular to the axis of the beam. Therefore, a beam experiences bending resulting in deflection in the y direction, which is a function of x. 8 A simply supported beam. 9 Euler–Bernoulli assumption. 1 Stress and Strain The stresses on the cross section of a beam are normal stress σxx and shear σxy .

4 Stresses on an infinitesimal cube. 4 shows an infinite small block in a solid. 13) where the term on the right-hand side is the initial force term, and bx is the component in the x direction of the external body force applied at the center of the small block. 20) This is the general form of dynamic equilibrium equation for elastic solids. 20. 21) Equations obtained in this section are applicable to 3D elastic solids. Theoretically, these equations can be applied to all other types of structures, such as trusses, beams, plates, and shells, because they are all made of 3D solids.