Definition

Numerical integration points, often referred to as Gauss points in the context of finite element analysis (FEA), are specific locations within an element where calculations are performed to approximate integrals of functions over the element. These points are used to accurately compute quantities such as stiffness matrices, force vectors, and stress distributions, especially in complex geometries where analytical integration is not feasible.

Calculation

Numerical integration points are determined based on the chosen integration method, such as Gaussian quadrature. The integration is performed by summing the weighted values of the function at these points. The basic formula for Gaussian quadrature in one dimension is:

∫ f(x) dx ≈ Σ w_i * f(x_i)

where:

• w_i are the weights associated with each integration point x_i, and
• x_i are the numerical integration points within the element.

Context of Use

Numerical integration points are crucial in various applications within geotechnical and structural engineering, including:

• Finite Element Analysis (FEA): Numerical integration points are used in FEA to compute element matrices and vectors, such as stiffness matrices and load vectors, ensuring accurate results in simulations.
• Stress and Strain Computation: In stress analysis, integration points are used to evaluate the stress and strain at specific locations within an element, which are then extrapolated to determine the overall behavior of the structure.
• Nonlinear Material Modeling: In nonlinear material models, numerical integration points help capture the complex behavior of materials under varying loads by evaluating the material response at different points within the element.
• Dynamic Analysis: Integration points are used in dynamic analysis to accurately capture the response of a structure to time-varying loads, such as seismic or wind loads.

Advantages

• Accurate Approximation: Numerical integration points provide a highly accurate approximation of integrals, essential for reliable FEA results.
• Efficiency: Using a finite number of integration points reduces computational complexity while maintaining accuracy, making it feasible to analyze large and complex models.

Limitations

• Dependent on Element Type: The choice and effectiveness of numerical integration points depend on the element type and the complexity of the geometry, requiring careful selection for optimal results.
• Potential for Inaccuracy in Highly Nonlinear Problems: In highly nonlinear problems, a limited number of integration points may lead to inaccuracies, necessitating a higher number of points or alternative integration methods.

Summary

Numerical integration points are a fundamental aspect of finite element analysis, enabling the accurate approximation of integrals within elements. These points are critical for computing essential quantities like stiffness matrices, force vectors, and stress distributions in complex geometries and material models. While they offer significant advantages in terms of accuracy and computational efficiency, their selection must be tailored to the specific element type and problem complexity to ensure reliable results.