Section 4: Implementation of Finite Element Analysis – Other Elements Quadrilate

Section 4: Implementation of Finite Element Analysis – Other Elements Quadrilate www.phwiki.com

Section 4: Implementation of Finite Element Analysis – Other Elements Quadrilate

Crawford-Whitehead, Estelle, Managing Editor has reference to this Academic Journal, PHwiki organized this Journal Section 4: Implementation of Finite Element Analysis – Other Elements Quadrilateral Elements Higher Order Triangular Elements Isoparametric Elements Section 4.1: Quadrilateral Elements Refers in general to any four-sided, 2D element. We will start by considering rectangular elements with sides parallel to coordinate axes. (Thickness = h) 4.1: Quadrilateral Elements (cont.) Normalized Element Geometry – “St in addition to ard” setting as long as calculations: Mapping between real in addition to normalized coordinates:

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4.1: Quadrilateral Elements (cont.) First Order Rectangular Element (Bilinear Quad): 4 nodes; 2 translational d.o.f. per node. Displacements interpolated as follows: “Bilinear terms” – implies that all shape functions are products of linear functions of x in addition to y. 4.1: Quadrilateral Elements (cont.) Shape Functions: 4.1: Quadrilateral Elements (cont.) Displacement interpolation becomes: Need to compute [B] matrix:

4.1: Quadrilateral Elements (cont.) Chain rule: Resulting [B(x)] matrix: Recall general expression as long as [k]: 0 0 Express in terms of in addition to ! 4.1: Quadrilateral Elements (cont.) Can show that Can also show that Everything in terms of in addition to ! 4.1: Quadrilateral Elements (cont.) Gauss Quadrature: Let’s take a closer look at one of the integrals as long as the element stiffness matrix (assume plane stress): Can be solved exactly, but as long as various reasons FEA prefers to evaluate integrals like this approximately: Historically, considered more efficient in addition to reduced coding errors. Only possible approach as long as isoparametric elements. Can actually improve per as long as mance in certain cases!

4.1: Quadrilateral Elements (cont.) Gauss Quadrature: Idea: approximate integral by a sum of function values at predetermined points with optimal weights – n = order of quadrature; determines accuracy of integral. (Note: any polynomial of order 2n-1 can be integrated exactly using nth order Gauss quadrature.) weights = known constants, depend on n Gauss points = known locations, depend on n 4.1: Quadrilateral Elements (cont.) Gauss Quadrature: Have tables as long as weights in addition to Gauss points: 2D case h in addition to led as two 1D cases: 4.1: Quadrilateral Elements (cont.) Higher Order Rectangular Elements More nodes; still 2 translational d.o.f. per node. “Higher order” higher degree of complete polynomial contained in displacement approximations. Two general “families” of such elements: Serendipity Lagrangian

4.1: Quadrilateral Elements (cont.) Lagrangian Elements: Order n element has (n+1)2 nodes arranged in square-symmetric pattern – requires internal nodes. Shape functions are products of nth order polynomials in each direction. (“biquadratic”, “bicubic”, ) Bilinear quad is a Lagrangian element of order n = 1. 4.1: Quadrilateral Elements (cont.) Lagrangian Shape Functions: Uses a procedure that automatically satisfies the Kronecker delta property as long as shape functions. Consider 1D example of 6 points; want function = 1 at in addition to function = 0 at other designated points: 4.1: Quadrilateral Elements (cont.) Lagrangian Shape Functions: Can per as long as m this as long as any number of points at any designated locations. No -k term! Lagrange polynomial of order m at node k

4.1: Quadrilateral Elements (cont.) Lagrangian Shape Functions: Use this procedure in two directions at each node: 4.1: Quadrilateral Elements (cont.) Notes on Lagrangian Elements: Once shape functions have been identified, there are no procedural differences in the as long as mulation of higher order quadrilateral elements in addition to the bilinear quad. Pascal’s triangle as long as the Lagrangian quadrilateral elements: 3 x 3 n x n 4.1: Quadrilateral Elements (cont.) Serendipity Elements: In general, only boundary nodes – avoids internal ones. Not as accurate as Lagrangian elements. However, more efficient than Lagrangian elements in addition to avoids certain types of instabilities.

4.1: Quadrilateral Elements (cont.) Serendipity Shape Functions: Shape functions as long as mid-side nodes are products of an nth order polynomial parallel to side in addition to a linear function perpendicular to the side. E.g., quadratic serendipity element: 4.1: Quadrilateral Elements (cont.) Shape functions as long as corner nodes are modifications of the shape functions of the bilinear quad. Step 1: start with appropriate bilinear quad shape function, . Step 2: subtract out mid-side shape function N5 with appropriate weight Step 3: repeat Step 2 using mid-side shape function N8 in addition to weight 4.1: Quadrilateral Elements (cont.) Notes on Serendipity Elements: Once shape functions have been identified, there are no procedural differences in the as long as mulation of higher order quadrilateral elements in addition to the bilinear quad. Pascal’s triangle as long as the serendipity quadrilateral elements: 3 x 3 m x m

4.1: Quadrilateral Elements (cont.) Zero-Energy Modes (Mechanisms; Kinematic Modes) – Instabilities as long as an element (or group of elements) that produce de as long as mation without any strain energy. Typically caused by using an inappropriately low order of Gauss quadrature. If present, will dominate the de as long as mation pattern. Can occur as long as all 2D elements except the CST. 4.1: Quadrilateral Elements (cont.) Zero-Energy Modes – De as long as mation modes as long as a bilinear quad: 1, 2, 3 = rigid body modes; can be eliminated by proper constraints. 4, 5, 6 = constant strain modes; always have nonzero strain energy. 7, 8 = bending modes; produce zero strain at origin. 4.1: Quadrilateral Elements (cont.) Zero-Energy Modes – Mesh instability as long as bilinear quads using order 1 quadrature: “Hourglass modes”

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4.1: Quadrilateral Elements (cont.) Zero-Energy Modes – Element instability as long as quadratic quadrilaterals using 2×2 Gauss quadrature: “Hourglass modes” 4.1: Quadrilateral Elements (cont.) Zero-Energy Modes – How can you prevent this Use higher order Gauss quadrature in as long as mulation. Can artificially “stiffen” zero-energy modes via penalty functions. Avoid elements with known instabilities! Section 4: Implementation of Finite Element Analysis – Other Elements Quadrilateral Elements Isoparametric Elements Higher Order Triangular Elements Note: any type of geometry can be used as long as isoparametric elements; we will only look at quadrilateral elements.

Section 4.2: Isoparametric Elements For various reasons, need elements that do not “fit” the st in addition to ard geometry. Curved boundaries Transition regions 4.2: Isoparametric Elements (cont.) Problem: How do you map a general quadrilateral onto the normalized geometry 4.2: Isoparametric Elements (cont.) Idea: Approximate the mapping using “shape functions”. Require to have Kronecker delta property. not required to be the actual shape functions of the element; n can be as large or as small as you want.

4.2: Isoparametric Elements (cont.) Solution: 2 x 2 Gauss quadrature: 4.2: Isoparametric Elements (cont.) Solution: Element nodal as long as ces:

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