How are the integration points of a quadrature determined?

How are the integration points of a quadrature determined?

The integration points and weights for the first orders of Gaussian quadrature in 1D are shown in the table below. The integral is taken over the normalized interval [-1,1]. In COMSOL Multiphysics, integration orders are designated by the degree of the polynomial that can be exactly integrated.

When to use order two Gauss rule for integration?

■Stresses calculated from are most accurate at Gauss points that were used for integration of the stiffness matrix. ■Usually an order two Gauss rule (four points) are used to integrate kof 4- and 8-node plane elements. ■Eight points are commonly used for 3-D elements.

Which is true about the Gaussian quadrature method?

The integration points are often called Gauss points, even though this nomenclature, strictly speaking, is correct only for integration points defined by the Gaussian quadrature method. In COMSOL Multiphysics, true Gaussian quadrature is used for integration in 1D, quadrilateral elements in 2D, and hexahedral elements in 3D.

What is the accuracy of Gauss point integration?

For true Gauss point integration, the accuracy is, as shown above, always an odd power. The integration order designated by the software as “4” will actually have accuracy 5 on those element shapes where Gaussian quadrature is used. Over the square -1 ≤ x ≤ 1, -1 ≤ y ≤ 1, the integral of this function is 1.

When to use true Gaussian quadrature In multiphysics?

In COMSOL Multiphysics, true Gaussian quadrature is used for integration in 1D, quadrilateral elements in 2D, and hexahedral elements in 3D. Other, similar schemes are used for other element geometries.

What is the Galerkin method for finite elements?

Galerkin approach for equations (1), (4), (5): 1. Multiply by weighting function w 2. Integrate over the domain 3. Discretize and sum the contributions of each element in domain 5/6/2015 Adrian Egger | FEM I | FS 2015 5 From Strong to Weak form II Apply the divergence theorem: Equivalent to integration by parts in 1D

Which is the default integration order in COMSOL Multiphysics?

The number describing the integration order in COMSOL Multiphysics is the highest order of the polynomial that can be integrated exactly. The default integration order is based on the order of the shape functions used to describe the field (in this case, the displacements).