J.E. Akin

Chapter 15 – SENSITIVITY ANALYSIS

Publisher Summary The optimal design of structural and thermal systems is usually obtained by employing finite element models. Usually, these are viewed as either a shape optimization for minimum weight or fully stressed designs or the determination of design parameters, such as thickness, for a given shape. Sensitivity analysis is an important aspect of these optimal design problems. Various sensitivity formulations are available but the question is which is computationally most efficient when used in conjunction with a finite element analysis. Most of the literature addresses structural problems. This chapter presents a review of the notation for finite element structural analysis. It introduces the notation for the optimal design problem, the sensitivity analysis for structures, and a new method for addressing stress failure criteria sensitivity.

Implementing FEA and CAD on personal computers and workstations

Abstract Currently, the use of the personal computer and the personal engineering workstation is rapidly increasing. However, many engineers are not aware of the power and flexibility that they offer. These systems can offer support for important engineering software such as finite element analysis (FEA) and computer aided design (CAD). This work will cover current capabilities in these areas, discuss their interaction, and predict some trends in capabilities for the personal computers.

TIME DEPENDENT PROBLEMS

This chapter provides an overview of time dependent problems. Many problems require the solution of time-dependent equations. In this context, there are numerous theoretical topics that an analyst should investigate before selecting a computational algorithm. These include the stability limits, amplitude error, and phase error. A large number of implicit and explicit procedures have been proposed. The chapter mentions typical additional computational procedures that arise in the time integration problems. It illustrates both simple explicit and implicit algorithms. Applications involving second-order spatial derivatives and first-order time derivatives will be referred to as parabolic or transient, while those with second-order time derivatives will be referred to as hyperbolic or dynamic. For these classes of problems, it is common to select various temporal operators to approximate the time derivatives. The actual time integration algorithm is determined by the choices of the difference operators and the ways that they are combined.

TRUSS ELEMENTS AND AXIS TRANSFORMATIONS

This chapter provides an overview of truss elements and axis transformations. The truss element is a very common structural member. A truss element is a two-force member, that is, it is loaded by two equal and opposite collinear forces. These two forces act along the line through the two connection points of the member. In elementary statics, the forces in truss elements are computed as if they were rigid bodies. However, there was a class of problems, called statically indeterminant, that could not be solved by treating the members as rigid bodies. With the finite element approach, both classes of problems can be solved. The elastic bar is a special form of a truss member. To extend the previous work to include trusses in two- or three-dimensions basically requires some review of analytic geometry. This chapter presents a review of that subject.

ELEMENT INTERPOLATION AND LOCAL COORDINATES

This chapter presents how the common interpolation functions are derived. It presents a number of expansions without proof. It also provides an overview of the concept of nondimensional local or element coordinate systems. The chapter discusses linear interpolation, quadratic interpolation, Lagrange interpolation, Hermitian interpolation, and hierarchical interpolation. The simplest interpolation would be linear and the simplest space is the line, for example, x -axis. Quadratic functions are completely different from the linear functions. Linear, quadratic, and Lagrange interpolation functions have C 0 continuity between elements. That is, the function being approximated is continuous among elements but its derivative is discontinuous. Some alternate types of interpolation have become popular. They are called hierarchical functions. The unique feature of these polynomials is that the higher order polynomials contain the lower order ones. The chapter also discusses interpolation error.

AUTOMATIC MESH GENERATION

This chapter describes the automatic mesh generation technique. The practical application of the finite element method requires extensive amounts of input data. Special mesh generation programs could help reduce this burden. The most powerful commercial codes offer extensive mesh generation and data supplementation options. Most mesh generation techniques are divided into two general classes: (1) automatic generation and (2) manual generation. In manual generation, the user at least determines how many elements will be used in each section of the mesh and how many element neighbors each will have. In the extreme case, the manual method means the analyst supplies all the nodal and element data. It is common for manual procedures to employ an isoparametric mapping or an IJK mapping. A typical mesh generation program locates and numbers the nodal points, numbers the elements, and determines the element incidences. In addition, it usually allows for assignment of codes to each node and each element. Thus, it also has a capability to generate boundary condition codes and element material codes. Many such codes use a mapping, or transformation, method to generate the mesh data. This chapter also illustrates some of the techniques for mesh generation.

CYLINDRICAL ANALYSIS PROBLEMS

There are many problems that can accurately be modeled as being revolved about an axis. Many of these can be analyzed by employing a radial coordinate, R , and an axial coordinate, Z . Solids of revolution can be formulated in terms of the two-dimensional area that is revolved about the axis. This chapter illustrates this concept. Numerous other objects are very long in the axial direction and can be treated as segments of a cylinder. This reduces the analysis to a one-dimensional study in the radial direction. The chapter also discusses that common special case. Changing to these cylindrical coordinates will make small changes in the governing differential equations and the corresponding integral theorems that govern the finite element formulation. Also the volume and surface integrals take on special forms.

MODEL APPLICATIONS IN 1-D

The simplest applications of the finite element method involve problems that are independent of time. This chapter discusses the programming of the problem-dependent subroutines associated with the application of interest. It presents several representative examples of the application of the finite element method to steady state problems. Generally, an element that can be defined by a single spatial variable in the local coordinates is relatively simple to implement. Thus, the typical illustrative applications begin with this class of element. This does not mean that the global coordinates must also be one-dimensional. For example, a set of one-dimensional truss elements can be utilized to construct a three-dimensional structure.

ERROR MEASURES FOR ELLIPTIC PROBLEMS

The Zienkiewicz and Zhu (or Z-Z) error estimator has become popular for loworder elements in adaptive applications. However, it has some serious problems, in general, that have been overcome by its recent extension. This chapter outlines the basic method and notation of these types of approaches for error estimation. It also outlines the Z-Z procedures to obtain continuous nodal strains or stresses to be utilized in the error norms. Some studies of the Z-Z method show that it is only reliable for quadratic elements applied to elliptic problems. For other elements, it does not act as a good error estimator. However, this approach can give smoothed values useful for post-processing contour displays.

Chapter 9 – GENERAL INTERPOLATION

Publisher Summary This chapter provides an overview of general interpolation. In a one-dimensional problem, it does not make a great deal of difference if one selects a local or global coordinate system for the interpolation equations, because the inter-element continuity requirements are relatively easy to satisfy. That is not true in higher dimensions. To obtain practical formulations, it is almost essential to utilize local coordinate interpolations. Doing this requires a small amount of additional work in relating the derivatives in the two coordinate systems. The chapter presents some of the procedures for deriving the interpolation functions in unit coordinates. It illustrates the Serendipity interpolation functions for quadrilateral elements that can be either, linear, quadratic, or cubic on any of its four sides. Such an element is often referred to as a transition element. The chapter also discusses hierarchical interpolation and quadrilateral elements or the quadrilateral faces of a solid element.