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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
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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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FE Mesh Construction Principles
Mesh Ill-conditioning
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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FE Mesh Construction Principles
Mesh Ill-conditioning
Element corner angle should be maintained
between 30 and 150
Distorted finite element can
increase the numerical error
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FE Mesh Construction Principles
Mesh Ill-conditioning
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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FE Mesh Construction Principles
Domain Extent Analysis
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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FE Mesh Construction Principles
Domain Extent Analysis
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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FE Mesh Construction Principles
Radially-graded Mesh
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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Radially-graded Mesh
FE Mesh Construction Principles
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
FE Mesh Construction Principles
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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
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