Previous class: Solution of FEM Linear Systemspaulino.ce.gatech.edu/courses/cee570/2014/Class_notes/...Previous class: Solution of FEM Linear Systems Sparse solvers (Direct & Iterative)

Previous class: Solution of FEM Linear Systems

Sparse solvers (Direct & Iterative)

Resequencing

Sparse Matrix terminology

CEE570 / CSE 551 Class #6/7

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3 Electronic Handouts

W. Celes, G.H. Paulino and R. Espinha “A compact adjacency-based topological data structure

finite element mesh representation, International Journal for Numerical Methods in Engineering,

Vol.64, No.11, pp.1529-1556, 2005

W. Celes, G.H. Paulino and R. Espinha “Efficient handling of implicit entities in reduced mesh

Representations, ASME Journal of Computing and Information Science in Engineering, Vol.5,

No.4, pp.348-359, 2005.

G. Li, G.H. Paulino and N.R. Aluru, Coupling of the mesh-free finite cloud method with the boundary

element method: a collocation approach, Computer Methods in Applied Mechanics and Engineering,

Vol.192, Nos.20-21, pp.2355-2375, 2003.

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Office Hours

Prof. Paulino

3129B Newmark Lab.

Thursdays, 3-5pm

Heng-Chi***, 1225 NCEL, Mondays 1-3pm

Shelly, 2310 Yeh, Tuesdays 1-3pm

Junho, 3310 Yeh, Wednesdays 1-3pm

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This class: Meshing Guidelines

FEM Terminology

FEM Mesh Construction Principles

Mesh Refinement

Mesh Ill-conditioning

Domain Extent Analysis

Infinite Elements

Radially-graded Mesh

Traditional Transition Mesh

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Meshing Guidelines

Edge Crack with Mesh Transition

Software I-Franc2D: Illinois FRacture ANalysis Code 2D Reference: Paulino and Kim “A new approach to compute T-stress in functionally graded materials

by means of the interaction integral method” Engineering Fracture Mechanics, Vol. 71, Nos. 13-14,

pp.1907-1950, 2004.

http://www.ghpaulino.com

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CEE570/CSE551

Meshing Guidelines

Welded Plates with Crack Software: FEACrack

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CEE570/CSE551

Meshing Guidelines

http://www.tecplot.com/showcase/gallery/mesh/thumbnails.htm

Propeller and Shaft

Tecplot Mesh Plot Gallery

http://www.tecplot.com/showcase/gallery/mesh/thumbnails.htm

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FEM Terminology

Element: A geometric sub-domain of the region

being simulated, with the property that it allows a

unique derivation of the approximation

(interpolation) functions.

Node: A geometric location in the element which

plays a role in the derivation of the interpolation

functions and is the point at which solution is

sought.

Mesh: A collection of elements (or nodes) that

replaces the actual domain.

Weak Form: An integral statement equivalent to

the governing equations and natural boundary

conditions.

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FEM Terminology (… continued)

Finite Element Model: A set of algebraic

equations relating the nodal values of the

PRIMARY VARIABLES (e.g. displacements) to the

nodal values of the SECONDARY VARIABLES

(e.g. forces) in an element.

Numerical Simulation: Evaluation of the

mathematical model (i.e. solution of the

governing equations) using a numerical method

and computer.

Important remark: Finite Element Model is NOT

the same as the Finite Element Method. There is

only one finite element method, but there can be

more than one finite element model of a problem

(or mathematical model).

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Paper: Topological Data Structure

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Paper: Topological Data Structure

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FE Mesh Construction Principles

Mesh Refinement

Refined mesh is often constructed by dividing

an existing element by two for each coordinate

Mesh refinement improves the accuracy of

finite element solution

Mesh refinement always requires more

computational time

Therefore, the analyst should maintain

the balance between the mesh refinement

and computational time.

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FE Mesh Construction Principles Mesh Refinement

I

I II III IV

II III IV

Loading

PCC Slab

Base

Subbase

Soil Degrees of Mesh Refinement

Stress Stress Stress Stress

J. Kim “Three-dimensional finite element analysis of multi-layered systems”

PhD Thesis, CEE Dept., University of Illinois at Urbana-Champaign, 2000.

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Aspect ratio of finite element should be

maintained under 5:1 for the finite element

model subjected to the bending and shear

deformation

Problem size can be reduced

but shear locking could

produce numerical error

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Element corner angle should be maintained

between 30 and 150

Distorted finite element can

increase the numerical error

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Smooth transition must be achieved to capture

an accurate displacement field

Rapid transition cannot

capture accurate displacement

field in FE solution

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An infinite or infinitely large domain is often truncated and the artificial boundary condition is applied to reduce the problem size and computational time

Domain truncation point should be determined by the domain extent analysis

• Evaluates the effect of truncated domain with respect to the variation of field variable

Domain must be extended until the field variable shows convergence with respect to the change of domain size

• At least three domain sizes should be evaluated

Finite element mesh in the core part should be identical for every model tried for the domain extent analysis

• Otherwise, the displacement can be affected by the different finite element mesh

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Domain Extent in Load Radii

-0.025

-0.090 0 200 100 300

Exact

Vert

ical

Dis

pla

cem

en

t (c

m)

x

x

Sampling

point

J. Kim “Three-dimensional finite element analysis of multi-layered systems”

PhD Thesis, CEE Dept., University of Illinois at Urbana-Champaign, 2000.

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FE Mesh Construction Principles Infinite Elements

Infinite element can simulate the decay of displacement in the subgrade layer away from the loading

Infinite element can replace the artificial boundary condition applied after the domain truncation

• Able to obtain accurate displacement prediction with small domain size

Domain extent analysis is required to determine the size of finite element domain

• The decay of field variable is governed by the (1 / r)^n rules. If the displacement of core region does not depend on those functions, domain extent analysis is required

Infinity direction is defined by the nodal incidence (element and node connectivity table)

• Local node number 1 to 4 (for 3-D) or 1 to 2 (for 2-D) must

be located at the finite and infinite element interface and

5 to 8 (for 3-D) or 3 to 4 (for 2-D) must be located toward

infinity direction

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Provides smooth transition and minimize mesh

ill-conditioning problem in finite element

model

Easily and efficiently connects refined and

coarse mesh in a finite element model

Provides best three-way transition in the 3-D

mesh

Minimizes the problem size with reasonable

solution accuracy

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FE Mesh Construction Principles Radially-graded Mesh

Aspect ratio < 1.1

70º < Corner angle < 120º

A B

Area A/B < 1.2

Infinite elements

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Infinite elements

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Traditional Transition Mesh Traditional transition mesh is easy to construct but

may yield mesh ill-conditioning problem

Traditional transition should be used with intervals in

order to avoid sudden change of finite element mesh

density

It is very difficult to make a two-way transition in the

3-D space


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FE Model Construction with PATRAN

Mesh generation tips

2-D mesh should be created in the xy plane

Use z coordinates to demonstrate depth in 3-D mesh generation

Estimate number of element by hand calculation …

Plan finite element mesh on the paper and test them in 2-D space before actual 3-D mesh creation

Avoid using “traditional mesh transition.” Employ smooth mesh transitions with well-graded elements.

Before solving a large-scale problem, try with small test problems and practice by yourself

Use right-hand-rule to create surface or solid

Solid or surface for the infinite element must be created toward infinity direction

Next class

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Variational methods

Exact vs. approximate solution

Principle of virtual work

Principle of stationary potential energy

Documents

Previous class: Solution of FEM Linear Systemspaulino.ce.gatech.edu/courses/cee570/2014/Class_notes/...Previous class: Solution of FEM Linear Systems Sparse solvers (Direct & Iterative)