Problem 54 - Concepts and Applications of Finite Element Analysis

Question

Let the following elements be rectangular in geometry, with side nodes evenly spaced and thicknesses constant. What order of Gauss quadrature is needed to obtain the exact stiffness matrix-that is, to integrate each \(k_{i j}\) exactly? (a) Plane bilinear element (four nodes). (b) Plane quadratic element (eight nodes). (c) Solid trilinear element (eight nodes). (d) Plane quadratic triangle (six nodes).

Step-by-Step Solution

Answer

(a) For a plane bilinear element (four nodes), 2 points are needed. (b) For a plane quadratic element (eight nodes), 3 points are needed. (c) For a solid trilinear element (eight nodes), 2 points are needed. (d) For a plane quadratic triangle (six nodes), 3 points are needed.

Step 1: Plane bilinear element (four nodes)

Considering the bilinear shape function, the number of quadrature points necessary to exactly evaluate the integrals for a plane is equal to the square root of the number of nodal points or shape functions. So for a quadrilateral element with 4 nodes, \( \sqrt{4} = 2 \) points are needed for Gaussian Quadrature.

Step 2: Plane quadratic element (eight nodes)

In a plane quadratic element with 8 nodes, the required number of Gauss quadrature points is the square root of the number of nodes or shape functions. So, \( \sqrt{8} \approx 2.83 \). Since the points in Gauss Quadrature are integers, we will round it up to the nearest integer. Hence, 3 points are needed.

Step 3: Solid trilinear element (eight nodes)

For the 3D trilinear element with 8 nodes, the number of quadrature points is the cube root of the number of nodes or shape functions. So, \( \sqrt[3]{8} = 2 \). Thus, 2 points are needed for Gaussian Quadrature.

Step 4: Plane quadratic triangle (six nodes)

For the plane quadratic triangle with 6 nodes, the required number of Gauss quadrature points is the square root of the number of nodes or shape functions. Therefore, \( \sqrt{6} \approx 2.449 \). Gauss quadrature points are integers, and therefore this will be rounded up to 3. Hence, 3 points will be needed.

Key Concepts

Gauss quadrature

Gauss quadrature is a numerical integration method used in finite element analysis that allows us to approximate integrals more efficiently. It's particularly useful in cases where we need to compute integrals over elements with complex shapes or where analytical integration would be complicated or impossible. Gauss quadrature involves selecting specific points within the element – known as Gauss points – and calculating the function's value at these points. Each point is assigned a weight, and the integral's approximate value is the sum of these weighted function values. The order of the Gauss quadrature determines the number of points used. For a function that is a polynomial of degree n, we typically need n/2 Gauss points to integrate it exactly. This method is highly efficient because it minimizes the number of evaluations necessary for a given accuracy. In practice:

• It is simple to implement, reducing the computational cost compared to other methods.
• Provides exact results for polynomial functions up to a certain degree, depending on the number of points chosen.
• Commonly used in finite element analysis for structural engineering, fluid dynamics, and more.

plane bilinear element

A plane bilinear element is a type of rectangular finite element with four nodes, generally utilized in two-dimensional problems. This element is defined by straight lines between each pair of connected nodes. In terms of geometry, it is a quadrilateral shape. These elements make use of bilinear shape functions. These shape functions can interpolate variables at the nodes throughout the element. Each of the element's sides is linear, allowing simple linear interpolation of displacement or any other field variable. For bilinear elements:

• Only 2 Gauss points in each direction are needed to perform an exact integration of linear functions over the element.
• They are beneficial for modeling materials and structures under uniform conditions.
• Have rapid computation due to fewer nodes and a simpler formulation than higher-order elements.

This simplicity and efficiency make them popular in finite element models to represent physical systems with fewer degrees of freedom.

plane quadratic element

Plane quadratic elements are a more advanced version of bilinear elements, commonly used in finite element analysis for 2D problems. These elements use eight nodes instead of four, which allows for parabolic rather than linear interpolation of variables across the element. With plane quadratic elements:

• More accurately capture variations within the element than plane bilinear elements.
• They require 3 Gauss points in each direction for exact integration of quadratic functions.
• Often used in stress analysis and dynamic simulations where higher precision is needed.

The key advantage this element provides is the ability to represent complex geometries and gradients in the physical properties more precisely than lower-order elements.

solid trilinear element

Solid trilinear elements represent a type of three-dimensional finite element with eight nodes positioned at the element's corners. These are crucial for representing solid materials in finite element models. Each node in these elements has three degrees of freedom, corresponding to the three-dimensional space. The term "trilinear" refers to the use of trilinear shape functions, which means that the interpolation of physical variables such as displacement within the element is linear concerning all three spatial coordinates. Key properties of solid trilinear elements include:

• Accurate representation of the physical structure in three dimensions with relatively simple computational efforts.
• Require only 2 Gauss points in each direction to achieve exact integration over the element.
• Widely used in structural engineering, geology, and other fields that involve 3D simulations.

Because they strike a balance between computational efficiency and accuracy, solid trilinear elements are ideal for creating realistic models without excessive computational resource demands.