Problem 2 - Concepts and Applications of Finite Element Analysis

Question

If element thickness can vary and is computed as \(t=\sum N_{i} t_{i}\) from nodal values \(t_{i}\), what order of Gauss quadrature is needed to compute the exact volume of (a) a four-node plane element, and (b) an eight-node plane element?

Step-by-Step Solution

Answer

The exact volume of a four-node plane element can be computed using Gauss quadrature of order 1, while for an eight-node plane element, a quadrature order of 2 is needed.

Step 1: Understanding relationship

To determine the order of the Gauss quadrature, it is crucial to understand the relationship between the polynomial order of the thickness distribution and the Gauss quadrature order. We must first note that the thickness \(t\) is expressed as a linear or bilinear combination (interpolation) of the nodal values \(t_{i}\). For a four-node element, \(t = N_1t_1 + N_2t_2 + N_3t_3 + N_4t_4\) where \(N_{i}\) are the linear shape functions. For an eight-node element, \(t\) is a bilinear interpolation of the nodal values and hence, a second order polynomial.

Step 2: Applying Gauss quadrature rule

The Gauss quadrature rule for 2D integrals (like quadrilateral elements) states that precise integration results are obtained when using an order equal to \(floor((p+1)/2)\), where p is the order of the polynomial. For a first order polynomial (like a four-node element), the Gauss quadrature of order 1 (1 Gauss point) will provide an exact result.

Step 3: Determining quadrature order for the eight-node element

For the eight-node element, \(t\) is a second order polynomial. Applying the Gauss quadrature rule for 2D integrals again, we find that an order of 2 (4 Gauss points) is needed to find the exact volume of the element.

Key Concepts

Gauss Quadrature

Gauss Quadrature is a numerical technique used for evaluating integrals, especially in the context of finite element analysis. It plays a crucial role in accurately integrating functions over complex domains.
In essence, the method approximates the integral of a function by evaluating it at specific points called Gauss points. The selection of these points and their respective weights makes the approximation highly efficient and accurate.
For computing integrals over a plane element, the order of Gauss quadrature needed depends on the polynomial degree of the function being integrated. The rule ensures that a polynomial of degree up to, but not exceeding, twice the quadrature order minus one, is integrated exactly.

• This means, for a linear polynomial (degree 1), an order 1 Gauss quadrature is sufficient.
• For a quadratic polynomial (degree 2), you would need a quadrature order of 2.

This technique is particularly powerful in finite element analysis where functions are often complex polynomials.

Shape Functions

Shape Functions are mathematical constructs used in finite element analysis to interpolate the solution across the elements in the mesh.
They describe how the solution varies within an element and are pivotal in forming the stiffness matrix and force vector in FEM.
For a four-node plane element, linear shape functions are used, expressed as:

• \( N_i = a_i + b_i\xi + c_i\eta \)
• where \(\xi\) and \(\eta\) are the local coordinates of the element.

In contrast, eight-node elements use quadratic or bilinear shape functions. These allow for a more flexible and accurate representation of curved boundaries and variable material properties within the element.
Thus, through shape functions, the FEM solution can be tailored closely to the physical problem being analyzed.

Polynomial Interpolation

Polynomial Interpolation in the context of finite element methods is used to approximate complex functions using simpler polynomial equations.
In finite elements, interpolation is applied to construct approximate solutions over discretized domains.
The thickness of a plane element, for instance, can be expressed as a function of its nodal values through interpolation using shape functions.

• For a four-node element, the interpolation is linear, meaning that the thickness varies linearly across the element.
• For an eight-node element, the interpolation can be quadratic, providing greater accuracy in representing changes in thickness.

Effective polynomial interpolation ensures that the solution within the elements is both accurate and computationally efficient, crucial for obtaining reliable results in simulations.

Plane Element

A Plane Element is a fundamental component in the finite element analysis of structures like slabs, panels, and membranes.
These elements are used to model problems in two dimensions, reducing the computational effort needed compared to 3D models.
Plane elements can vary in complexity, with four-node and eight-node elements being quite common:

• Four-node plane elements are simpler and utilize linear shape functions, ideal for modeling structures with uniform properties.
• Eight-node plane elements are more complex, using quadratic shape functions to capture detailed variations over the element, like curvatures or variable material properties.

These elements are essential for analyzing how loads are distributed and how structures will deform, providing insights that influence design decisions across engineering disciplines.