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ECIV 720 A Advanced Structural Mechanics and Analysis
Lecture 16 & 17: Higher Order Elements (review) 3-D Volume Elements Convergence Requirements Element Quality
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Higher Order Elements Quadrilateral Elements Recall the 4-node Complete Polynomial 4 Boundary Conditions for admissible displacements 4 generalized displacements ai
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Assume Complete Quadratic Polynomial
Higher Order Elements Quadrilateral Elements Assume Complete Quadratic Polynomial 9 generalized displacements ai 9 BC for admissible displacements
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BT18x3 D3x3 B3x18 ke 18x18 9-node quadrilateral
9-nodes x 2dof/node = 18 dof BT18x3 D3x3 B3x18 ke 18x18
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9-node element Shape Functions
Following the standard procedure the shape functions are derived as 1 2 3 4 Corner Nodes x h 5 6 7 8 Mid-Side Nodes 9 Middle Node
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N1,2,3,4 Graphical Representation
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N5,6,7,8 Graphical Representation
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N9 Graphical Representation
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Polynomials & the Pascal Triangle
Degree 1 x y 1 2 x2 xy y2 3 x3 x2y xy2 y3 4 x4 x3y x2y2 xy3 y4 5 x5 x4y x3y2 x2y3 xy4 y5 …….
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Polynomials & the Pascal Triangle
To construct a complete polynomial Q1 4-node Quad 1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y x2y2 xy3 y4 ……. x5 x4y x3y2 x2y3 xy4 y5 Q2 9-node Quad etc
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Incomplete Polynomials
1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y x2y2 xy3 y4 ……. x5 x4y x3y2 x2y3 xy4 y5 3-node triangular
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Incomplete Polynomials
1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y x2y2 xy3 y4 ……. x5 x4y x3y2 x2y3 xy4 y5
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8 coefficients to determine for admissible displ.
8-node quadrilateral Assume interpolation 1 2 3 4 x h 5 6 7 8 8 coefficients to determine for admissible displ.
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BT16x3 D3x3 B3x16 ke 16x16 8-node quadrilateral
8-nodes x 2dof/node = 16 dof BT16x3 D3x3 B3x16 ke 16x16
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8-node element Shape Functions
Following the standard procedure the shape functions are derived as 1 2 3 4 Corner Nodes h 5 6 7 8 Mid-Side Nodes x
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N1,2,3,4 Graphical Representation
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N5,6,7,8 Graphical Representation
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Incomplete Polynomials
1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y x2y2 xy3 y4 ……. x5 x4y x3y2 x2y3 xy4 y5
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6 coefficients to determine for admissible displ.
6-node Triangular Assume interpolation 1 2 3 4 5 6 6 coefficients to determine for admissible displ.
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BT12x3 D3x3 B3x12 ke 12x12 6-node triangular
6-nodes x 2dof/node = 12 dof 1 2 3 4 5 6 BT12x3 D3x3 B3x12 ke 12x12
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6-node element Shape Functions
Following the standard procedure the shape functions are derived as Corner Nodes 1 2 3 Mid-Side Nodes 4 5 6 Li:Area coordinates
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Other Higher Order Elements
1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y x2y2 xy3 y4 ……. x5 x4y x3y2 x2y3 xy4 y5 12-node quad x h 1 2 3 4
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Other Higher Order Elements
1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y x2y2 xy3 y4 ……. x3y2 16-node quad x h 1 2 3 4 x5 x4y x3y2 x2y3 xy4 y5
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3-D Stress state
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3-D Stress State Assumption Small Deformations
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Strain Displacement Relationships
Material Matrix
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3-D Finite Element Analysis
Solution Domain is VOLUME 10 11 12 7 8 9 1 2 3 4 5 6 Simplest Element (Lowest Order) Tetrahedral Element
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3-D Tetrahedral Element
z 3 (0,0,1) x h Parent (Master) 1 (1,0,0) 2 (0,1,0) 4 (0,0,0) Can be thought of an extension of the 2D CST
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1 2 3 4 3-D Tetrahedral Shape Functions x h z Volume Coordinates
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Geometry – Isoparametric Formulation
In view of shape functions
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Jacobian of Transformation
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Strain-Displacement Matrix
B is CONSTANT
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Stiffness Matrix Element Strain Energy
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Force Terms Body Forces
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Element Forces Surface Traction Applied on FACE of element 1 2 3 4 eg on face 123
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se = DB qe Stress Calculations Stress Tensor Constant
Stress Invariants
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Stress Calculations Principal Stresses
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Other Low Order Elements
1 2 3 5 6 7 8 24 dof 6-hedral 1 2 3 4 5 6 18 dof 5-hedral
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Degenerate Elements 8 1 2 4 5 6 8 ,3 ,7 7 5 6 1 3 2 Still has 24 dof
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Degenerate 1 2 3 5 6 7 8 1 2,3 4 5,6,7,8 Still has 24 dof
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Higher Order Elements 10-node 4-hedral
2 3 4 5 6 7 8 9 10 3 4 6 8 10 1 9 7 13 14 15 5 Z 2 Y X
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15-node 5-hedral 3 2 4 6 8 10 11 12 13 14 15 1 9 7 5 L 1 3 z 2 4 5 6 7 8 9 10 11 12 13 14 15 Z Y X
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15-node 5-hedral Shape Functions
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20-node 6-hedral z 24 23 8 16 15 22 5 20 7 x h 13 14 6 17 4 19 12 11 18 1 3 Z 9 10 2 Y X
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20-node 6-hedral Shape Functions
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Convergence Considerations
For monotonic convergence of solution Requirements Elements (mesh) must be compatible Elements must be complete
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Monotonic Convergence
FEM Solution Exact Solution No of Elements For monotonic convergence the elements must be complete and the mesh must be compatible
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Mixed Order Elements Consider the following Mesh 4-node 8-node Incompatible Elements…
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Mixed Order Elements We can derive a mixed order element for grading 8-node 4-node 7-node By blending shape functions appropriately
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Convergence Considerations
For monotonic convergence of solution Requirements Elements (mesh) must be compatible Elements must be complete
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Element Completeness For an element to be complete Assumption for displacement field must accommodate RIGID BODY MOTION CONSTANT STRAIN STATE
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Element Completeness Consider This is not a complete polynomial However,
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Assume displacement field
Element Completeness Assume displacement field The Computed nodal displacement corresponding to this field i=1,…,#of nodes Test for ELEMENT completeness Isoparametric Formulation
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Element Completeness Isoparametric Formulation Thus, computed displacement field
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Element Completeness Computed Assumed In order for the computed displacements to be the assumed ones we must satisfy Condition for element completeness
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Effects of Element Distortion
Loss of predictive capability of isoparametric element No Distortion 1 x y x2 xy y2 x2y xy2 Behavior accurately predicted
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Effects of Element Distortion
Angular Distortion 1 x y x2 xy y2 x2y xy2 Predictability is lost for all quadratic terms
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Effects of Element Distortion
Quadratic Curved Edge Distortion 1 x y x2 xy y2 x2y xy2 Predictability is lost for all quadratic terms
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Effects of Element Distortion
The advantage (reduced #of dof) of using 8-node higher order element based on an incomplete polynomial is lost when high element distortions are present
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Effects of Element Distortion
Loss of predictive capability of isoparametric element No Distortion 9-node 1 x y x2 xy y2 x2y xy2 x2y2 Behavior accurately predicted
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Effects of Element Distortion
9-node Angular Distortion 1 x y x2 xy y2 x2y xy2 Behavior predicted better than 8-node
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Effects of Element Distortion
Quadratic Curved Edge Distortion 9-node 1 x y x2 xy y2 x2y xy2 Predictability is lost for high order terms
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Effects of Element Distortion
The advantage (reduced #of dof) of using higher order element based on an incomplete polynomial is lost when high element distortions are present For angular distortion 9-node element shows better behavior For Curved edge distortion all elements give low order prediction
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Polynomial Element Predictability
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Tests of Element Quality
Eigenvalue Test Identify Element Deficiencies Patch Test Convergence of Solutions
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Eigenvalue Test 1 2 3 4 Apply loads –{r} in proportion to displacements
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Eigenvalue Test Eigenproblem As many eigenvalues l as dof For each l there is a solution for {d}
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Displacement Modes & Stiffness Matrix
For all eigenvalues and modes
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Eigenvalue Test Scale {d} so that then
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Eigenvalue Test Rigid Body Motion => System is not strained => U=0 System is strained => U=0
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+ Rigid Body Motion Rigid Body Modes Element Straining Modes
Total Number of Element Displacement Modes (=number of degrees of freedom)
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Displacement Modes & Stiffness Matrix
Consider the 2-node axial element Identify all possible modes of displacement
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Displacement Modes & Stiffness Matrix
Consider the 4-node plane stress element 1 t=1 E=1 v=0.3 1 8 degrees of freedom 8 modes Solve Eigenproblem
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Displacement Modes & Stiffness Matrix
Rigid Body Mode Rigid Body Mode
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Displacement Modes & Stiffness Matrix
Rigid Body Mode
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Displacement Modes & Stiffness Matrix
Flexural Mode Flexural Mode
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Displacement Modes & Stiffness Matrix
Shear Mode
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Displacement Modes & Stiffness Matrix
Stretching Mode Uniform Extension Mode (breathing)
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Displacement Modes & Stiffness Matrix
The eigenvalues of the stiffness matrix display directly how stiff the element is in the corresponding displacement mode
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Patch Test Objective Examine solution convergence for displacements, stresses and strains in a particular element type with mesh refinement
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Patch Test - Procedure Build a simple FE model Consists of a Patch of Elements At least one internal node Load by nodal equivalent forces consistent with state of constant stress Internal Node is unloaded and unsupported
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Patch Test - Procedure Compute results of FE patch If (computed sx) = (assumed sx) test passed
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