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1
THE ACCURACY OF FEA
2
REALITY
MATHEMATICAL
MODEL
FEA
MODEL
RESULTS
Discretization error
Modeling error
Solution error
Discretization error is
controlled in the
convergence process
3
DISCRETIZATION OF STRESS DISTRIBUTION
Mesh built with first order triangular elements called constant stress triangles
First order element assumes linear distribution of displacements within each element. Strain, being derivative of
displacement, is constant within each element. Stress is also constant because it is calculated based on strain.
Discrete stress distribution in constant stress triangles
4
Tensile hollow strip modelled with a coarse mesh of 2D plate elements.
An isometric view of von Mises effective stress distribution in
the upper right quarter of the model shown above. The
height of bars represents the magnitude of stress. Notice
that stresses are constant within each element.
DISCRETIZATION OF STRESS DISTRIBUTION
5
CONVERGENCE ANALYSIS BY MESH REFINEMENT
The same tensile strip modelled three times with increasingly refined meshes. The
process of pprogressive mesh refinement is called h convergence
6
CHARACTERISTIC ELEMENT SIZE
The process of pprogressive mesh refinement is called h convergence because
characteristic element size h is modified during this process
7
Discretization errors
Discretization error is an inherent part of FEA. It is the price we pay for discretization of a continuous structure.
Discretization error can be defined either as solution error or convergence error.
Convergence error
Convergence error is the difference between two consecutive mesh refinements and/or element order upgrade.
Let’s say convergence error is 10%. If convergence takes place, then the next refinement and/or element order
upgrade will produce results that will be different from the current one by less than 10%.
Solution error
The solution error is the difference between the results produced by a discrete model with a finite number of
elements and the results that would be produced by a hypothetical model with an infinite number of infinitesimal
elements. To estimate the solution error, one has to assess the rate of convergence and predict changes in
results within the next few iterations as if they were performed.
CONVERGENCE CURVE
1 2 3
MESH REFINEMENT AND / OR ELEMENT ORDER UPGRADE NUMBER
CO
NV
ER
GE
NC
E C
RIT
ER
ION
SOLUTION OF THE HYPOTHETICAL “INFINITE” FINITE ELEMENT MODEL (UNKNOWN)
SOLUTION ERROR FOR MODEL # 3
CONVERGENCE ERROR
FOR MODEL # 3
# OF D.O.F. IN THE MODEL
8
CONVERGENCE PROCESS WITH h ELEMENTS AND p ELEMENTS
The name h comes from characteristic element size usually denoted as h.
That characteristic element size is reduced during h convergence process.
The name p comes from polynomial function describing displacement field in the element.
The order of polynomial function is increased during p convergence process.
Both in h and p elements convergence process means adding degrees of freedom the the model. With h elements this is accomplished by mesh refinement. With p elements degrees of freedom and added by increasing elements order while mesh remains unchanged.
h - elements p - elements
9
h - elements p - elements
Element shape: tetrahedral, wedge, hexahedralElement shape: tetrahedral, wedge, hexahedral
Mapping allows for only little deviation from the ideal shape.
Displacement field mapped by lower order polynomials (1st or 2nd), polynomial order does not change during solution
Mapping allows for higher deviation from the ideal shape but may introduce errors on highly curved edges and surfaces
Displacement field described by mapped higher order polynomials (up to 9th ), polynomial order adjusted automatically to meet user’s accuracy requirements.
COMPARISON BETWEEN h ELEMENTS AND p ELEMENTS
results are produced in the iterative process that continues until the known, user specified accuracy, has been obtained
results are produced in one single run with unknown accuracy
fewer large elements typically 500 - 10000many small elements typically 5000 - 500000
Note: Only tetrahedral elements elements can be reliably created with the available automeshers
10
ec001 HOLLOW TENSILE STRIP
model file ec001
model type solid
material alloy steel
restraints built-in to left edge
load 100,000N tensile load to right edge in x direction
objectives
• meshing solid CAD geometry
• using solid elements
• demonstrating h convergence process
100,000 Ntensile load
Built-insupport
11
ec001 HOLLOW TENSILE STRIP
Step 1 of h convergence process
Maximum von Mises stress 345.3 MPaSecond order (high quality) solid elements used
12
ec001 HOLLOW TENSILE STRIP
Second order (high quality) solid elements used
Step 2 of h convergence process
Maximum von Mises stress 367.7 MPa
13
Step 3 of h convergence process
Maximum von Mises stress 375.3 MPa
ec001 HOLLOW TENSILE STRIP
Second order (high quality) solid elements used
14
ARTIFICIAL STIFNESS
ELEMENT STRESSES
NODAL STRESSES
H – ADAPTIVE SOLUTION
15
0
100
200
300
400
0 1 2 3 4
von
Mis
es
stre
ss [
MP
a]
ec001 HOLLOW TENSILE STRIP
345 MPa 368 MPa 375 MPaMPa
TDW
P167
10)40100(
000,100
)(
216.2100
401212
33
W
DKn
MPaK nn 370216.2167max
368 3700.5%
368
COSMOS THEORYerror
COSMOS
Results of h convergence process presented in the form of a convergence curve
Verification of stress results:Haywood formula, Peterson p. 111
Ratio of global element size at iteration # 1to current global element size