FEA 30: 2-D Gaussian Quadrature

Summary

This video script explains Gaussian quadrature in two dimensions, focusing on its application in finding the stiffness matrix for 2D quadrilateral elements in finite element analysis. It covers single and multiple integration points, illustrating how to use reduced and full integration with examples. The script also discusses the impact of integration methods on structural stiffness, noting that reduced integration can soften the structure.

Takeaways

• 📐 Gaussian quadrature is extended to two-dimensional elements for numerical integration.
• 🔍 The concept of single-point integration is introduced, simplifying the process by evaluating integrals at a single point in both directions.
• 📏 The standard element used is a bilinear quadrilateral element, ranging from -1 to 1 in each direction with an interval width of 2.
• 🧮 Reduced integration for a linear or Q4 element is explained, which involves evaluating the function at the center of the S&T coordinate system.
• 🔢 Full integration for a linear quadrilateral element (Q4) is achieved with 2x2 integration points, totaling 4 points.
• 🔄 The process is further expanded to 3x3 integration points for quadratic quadrilateral elements (Q8), providing exact stiffness matrix results for well-shaped elements.
• 📑 The integral to resolve involves the BDB terms inside the stiffness matrix expression, which is approximated as a double sum of weights times function evaluations.
• 📐 The locations for integration points are detailed, including the use of square root values to determine positions in the S and T directions.
• 🔑 The stiffness matrix for a Q4 element is found using full integration, which involves evaluating the integrand at multiple points and considering the Jacobian determinant.
• 🔧 The difference between full and reduced integration is highlighted, with full integration providing a stiffer structure and reduced integration potentially softening it.

Q & A

What is Gaussian quadrature and how is it extended to two-dimensional elements?

Gaussian quadrature is a method for numerical integration, which approximates the integral of a function by using weighted sum of the function's values at specified points. In two-dimensional elements, this concept is extended by using a double sum over the number of intervals in two directions, typically denoted as 's' and 't', to evaluate the integrals at multiple points in both directions.

What is the significance of using a single integration point in Gaussian quadrature for 2D elements?

Using a single integration point simplifies the calculation by reducing the number of evaluations needed. It's often referred to as reduced integration and is used for linear or bilinear quadrilateral elements. It evaluates the function at the center of the 's' and 't' coordinate system, which is when both 's' and 't' are equal to zero.

How does increasing the number of integration points affect the accuracy of the stiffness matrix calculation?

Increasing the number of integration points, such as moving from a 1x1 to a 2x2 or 3x3 grid, increases the accuracy of the stiffness matrix calculation. A 2x2 grid provides full integration for a linear quadrilateral element, while a 3x3 grid gives full integration for a quadratic element, potentially yielding exact results for the stiffness matrix.

What is the term 'reduced integration' in the context of the script?

Reduced integration refers to the practice of using fewer integration points than the full number required for exact integration. This approach can be used to reduce computational cost and sometimes to mitigate issues with overly stiff elements in finite element analysis.

Can you explain the term 'stiffness matrix' as mentioned in the script?

The stiffness matrix is a key component in finite element analysis that represents the structural stiffness of an element. It relates the nodal forces to the nodal displacements and is used to model the elastic properties of the element.

What is the role of the Jacobian determinant in the stiffness matrix calculation?

The Jacobian determinant is crucial in transforming the integral from the natural coordinate system ('s' and 't') to the global coordinate system. It accounts for the area transformation between these coordinate systems and is used to scale the integration weights correctly.

What does the 'B' matrix represent in the context of the stiffness matrix calculation?

The 'B' matrix is derived from the partial derivatives of the shape functions with respect to the coordinates ('s' and 't'). It relates the strains in the element to the nodal displacements and is a part of the stiffness matrix formulation.

How does the shape of the element affect the choice between full and reduced integration?

The choice between full and reduced integration can depend on the element's shape. For elements with a good shape, full integration can provide exact results. However, for elements with poor shape quality, reduced integration might be used to avoid numerical issues such as locking phenomena.

What are the implications of using reduced integration on the structural model's stiffness?

Using reduced integration can result in a softer structural model because it effectively reduces the stiffness of the elements. This can be beneficial in some cases to offset the natural tendency of finite element models to be overly stiff.

What is the significance of the 'D' matrix in the stiffness matrix calculation?

The 'D' matrix, also known as the material matrix, relates the stress and strain in the element according to the material's constitutive law. It is dependent on material properties such as Young's modulus and Poisson's ratio and is independent of the element's geometry.

How does the script suggest evaluating the stiffness matrix for a bilinear quadrilateral element?

The script suggests evaluating the stiffness matrix for a bilinear quadrilateral element using reduced integration by evaluating the 'B' matrix and the Jacobian determinant at the center of the 's' and 't' coordinate system (where 's' and 't' are both zero).