The Pennsylvania State University The Graduate School Department of Mechanical Engineering ANISOTROPIC H-ADAPTIVE (AH-ADAPTIVE) FINITE ELEMENT SCHEME FOR THREE-DIMENSIONAL MULTI-SCALE ANALYSES A Thesis in Mechanical Engineering by Shih-Horng Tsau c 2006 Shih-Horng Tsau Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2006
164
Embed
ANISOTROPIC H-ADAPTIVE (AH-ADAPTIVE) FINITE ELEMENT …
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Submitted in Partial Fulfillmentof the Requirements
for the Degree of
Doctor of Philosophy
December 2006
The thesis of Shih-Horng Tsau has been reviewed and approved* by the following: Panagiotis Michaleris Associate Professor of Mechanical Engineering Thesis Adviser Chair of Committee Ashok D. Belegundu Professor of Mechanical Engineering Eric Mockensturm Associate Professor of Mechanical Engineering Francesco Costanzo Associate Professor of Engineering Mechanics Karen A. Thole Professor of Mechanical Engineering Head of the Department of Mechanical Engineering *Signatures are on file in the Graduate School.
iii
Abstract
Using a static mesh in a multi-scale simulation, such as welding, requires many
fine elements from the start of the analysis. The mesh needs to be fine throughout the
entire simulation in both transverse and longitudinal directions to capture high gradients.
Isotropic adaptive meshing performs simultaneous coarsening, and refining, in all spatial
dimensions. Application of isotropic adaptive meshing allows the use of a coarse mesh
as the analysis begins and it refines as needed in all directions during the simulation.
However, because of the nature of isotropic refinement, elements need to remain fine in
all dimensions even if the gradient is high in only one direction. In this work, an effi-
cient Anisotropic h-Adaptive (abbreviated AH-adaptive) FEA method is developed that
performs independent refining and coarsening among all spatial dimensions. Application
of the anisotropic h-adaptive meshing allows the use of a coarse mesh as the analysis
starts. If there is one direction in which the gradient is much smoother than the others,
the mesh coarsens in the corresponding direction, thus reducing the number of DOFs by
n12 in 2D analyses, and n
13 in 3D analyses.
Dependent (also referred to as “constraint”) nodes occur when h-adaptive refine-
ment strategy is applied. The DOFs (degrees of freedom) on these dependent nodes must
be separated from the original system of algebraic equations. Only the unconstrained
(also referred to as “free”) DOFs can exist in the real equation system to solve and there-
fore yield accurate solution fileds. To deal with the dependent DOFs, several methods
can be applied for the numerical computations. A comparison between Condensation
iv
and Recovery Method, Lagrange Multiplier, and Penalty Method is performed. And the
Condensation and Recovery Method is chosen to be applied in the AH-adaptive FEA
scheme to maximize the computational efficiency.
Highlights from this research include important contributions such as: 1) simpli-
fied gradient calculations for each element, 2) nonzero fill-in effects induced by condensing
the original algebraic equation systems, 3) moving forced refinements of anticipated high
gradients, 4) procedures which assist meshes with neatly coarsening elements to the al-
lowed maximum, and 5) comparisons among possible approaches for the original system
equations from a mesh which has constrained nodes.
Note that the outer radius does not necessarily have to be much larger than the
inner radius for an analysis, because as long as the inner radius can generate sufficient
67
Table 3.2. Statistics of the AH-adaptive analysisCase number L1 L2Radii of EnforcedRefinement
12.14mm and 13.11mm 13.04mm and 14.14mm
Minimum Length ofElements
0.6mm 0.6mm
Permissible Gradient 20.0 20.0Total Number ofTime Increments
1144 1174
Number of Nodes Initial 440 Peak 14438 Initial 440 Peak 18502Number of Elements Initial 288 Peak 11193 Initial 299 Peak 13518Average number ofnodes per iteration
Save this convergentmesh(connectivities, solutions)
convergence efficiency improving strategy
Fig. 4.1. Flow chart of the AH-adaptive FE analysis scheme.
83
4.1.1 Small Deformation
In small deformation theory, the total strain ε is Green’s strain.
ε(r, t) =12
[∇ru(r, t) +
(∇ru(r, t)
)T]
(4.2)
Assuming small deformation thermo-elasto-plasticity, the total strain ε can be
decomposed into three terms:
ε = εe + εp + εt (4.3)
where εe, εp and εt are the elastic strain, plastic strain and thermal strain, respectively.
Using the above equation, the stress strain constitution relationship is
σ = C(ε − εp − εt) (4.4)
where C is the material stiffness matrix.
Applying the associative J2 plasticity [36], the yield function f is
f = σm
− σY
(εq, T ) (4.5)
where σm
and σY
are the Mises stress and yield stress.
Active yielding occurs when f ≥ 0. The evolution of εq
for active yielding can be
evaluated by the radial return algorithm [37], and then ε̇p
can be calculated from
ε̇p
= ε̇qa (4.6)
84
where a is the flow vector. The initial and boundary conditions can be found in [13].
4.1.2 Large Deformation
Illustrated in [38] is the large deformation theory:
For Total Lagrangian formulation, the basic equation is
∫V 0
St+�t
0,ijδε
t+�t
0,ijdV
0 = Rt+�t (4.7)
For Updated Lagrangian formulation, the basic equation is
∫V t
St+�t
t,ijδε
t+�t
t,ijdV
t = Rt+�t (4.8)
where
V0 is the volume at time 0
Vt is the volume at time t
St+�t
0,ij, S
t+�t
t,ijare the second Piola-Kirchhoff stress tensor
δεt+�t
0,ij, δε
t+�t
t,ijare the incremental Green-Lagrange strain tensor
Rt+�t is the external virtual work
An AH-adaptive mechanical analysis involves the tasks illustrated in the following
sections.
85
4.2 Initialization of Information Arrays
Throughout an entire analysis, the re-meshing procedures generate new entities.
Arrays which save this information associated with nodes and elements are necessary in
order to transfer the properties between the entities and properly construct the new mesh.
Figures 4.2 and 4.3 show the structures of the arrays. In mechanical analyses, element
arrays include Gauss point quantities (Figure 4.3), such as stresses and strains, which
are not used in thermal analyses [35]. Other than this difference, analogous concepts on
these node-wise and element-wise arrays are illustrated in [35].
4.3 Control Criteria on Generating Elements for Self-Adapting Dy-
namic Meshes
At the beginning of an analysis, and between two time increments, an initial/current
adaptive mesh will be modified to generate a new adaptive mesh by refining or coarsening
the elements as necessary. The key controlling criteria are:
1. Gradient measures. These gradients are compared to the desired permissible gradi-
ent. Elements are refined when the gradient measures are too high so as to request
for smaller element sizes, and coarsened if the measures are too low.
2. Moving forced refinement, which are set to track the moving heat source, and force
refinement in order to guarantee sufficient element density to adequately integrate
the heat source.
86
Node 1 Node 2 ..... Node Nmax
The temporary array for node data during re-meshing
x-coordinate
Nmax : the maximum # of total temporary nodes allowed during one loop of re-meshing
# of previous mesh nodes .....
y-coordinate
z-coordinate
nodal solution
*node-wise boundary conditions
Node 1 Node 2 .....
x-coordinate
# of active nodesafter re-meshing
y-coordinate
z-coordinate
nodal solution
*node-wise boundary conditions
The array of active node data
(With the information on what elements are active)
The nodes on active elements
* : actual number of rows depends on how many different kinds of boundary conditions may be applied in the analysis
Fig. 4.2. The information array of node data.
87
Whether it is active
The initial element it comes from
..... (1:Active)0
Emax : the maximum # of total temporary elements allowed during one loop of re-meshing
Element generations
Index number associated with the generations
Elem 1 Elem 2 ..... Elem Emax# of previous mesh elements .....
Surface-wise B.C.
Gauss point quantities
1 0 ..... 1
The temporary array for element data during re-meshing
Initial Elem 1 Intial Elem 2 ..... Initial Element N
Maximum generations to be refined(for all 3 local directions seperately)
The material group
Element type configuration
The base (reference) array for all initial elements
Elem 1 Elem 2 .....# of active elementsafter re-meshing
The array of active element data
: actual number of rows depends on how many different kinds of boundary conditions may be applied in the analysis
b
b
b
The initial element it comes from
Element generations
Index number associated with the generations
b
b
b
a
: contains the information rows for element type, number of nodes in an elementa
: contains 3 rows, for the information corresponding to r1,r2,r3-directionsb
d
c
c
d : contains rows for stresses, strains, equivalent plastic strain, etc.
Surface-wise B.C.
Gauss point quantitiesd
c
Remaining generationsto refine / coarsen
Remaining generationsto refine / coarsen
Fig. 4.3. The information array of element data.
88
4.4 Gradient Measure Definition
Except at the start of an analysis, the AH-adaptive scheme examines elements on
a current mesh after the solution field has been acquired. An element with a smoother
gradient field will tend to be given a larger element size through coarsening, while an
element with a higher gradient demands to be refined to a smaller size to better reflect
the steep gradient.
4.4.1 Review of isotropic norm definitions
For isotropic FE analyses, [25] applies error norm derivations which build on
concepts introduced in [26, 27]. The need to isotropically refine or coarsen the elements
in a mesh is calculated according to:
‖e(i)‖ ≤ Ce(i)
hp−m+d (4.9)
where e(i) denotes the error in an element i, h is the maximum element diagonal
length, p is the order of the shape function, m is the highest order of differentiation in
the strain-displacement relation, and d equals to 1, 2, or 3 depending on the number
of dimensions. A new element size is computed after the element local error norm is
normalized by an additionally evaluated global gradient field.
89
Though these norm equations serve for isotropic refinement and coarsening, an
anisotropic analysis requires calculations for gradients independently for all (local r1-,r2-
,r3-) directions. The gradient evaluation for the AH-adaptive FEA scheme is developed
as in the following section.
4.4.2 Gradient measures of AH-adaptive mechanical analysis
While in an AH-adaptive thermal analysis the gradient measures are evaluated
according to the temperatures in the previous mesh [35], in a mechanical analysis there
are many solution fields such as nodal displacements, element-wise plastic strains or
stresses, and nodal peak temperatures. These different quantities can be taken as the
basis of the gradient measures. The mechanical analyses which will be demonstrated are
structural responses simulations with inputs of temperature results (from an AH-adaptive
heat transfer analysis). Note that the AH-adaptive scheme can also be applied for a
pure mechanical simulation without temperature inputs, e.g. only subject to mechanical
forces.
Peak Temperature as the Gradient Measure
Take an example of using peak temperatures (the highest temperature a node of a
specific coordinates has experienced from the beginning up to a current time increment)
as the gradient measure. Consider that at a time increment, the peak temperatures
on the nodes of an element are expressed as Tp
= [Tp1, T
p2, ..., Tp8]T . Thus, the peak
temperature Tp
of any interior point in this element is calculated by
Tp
= N · Tp (4.10)
90
where N is the shape function. This leads to the gradient definitions of the
solution field in all directions as
∇rTp=
dTp
dr=
d(N · Tp)
dr=
dNdr
· Tp
(4.11)
where the local gradient ∇r is a 3 ∗ 1 vector.
The AH-adaptive scheme evaluates the local gradients in an element at the center
of isoparametric coordinates (r1, r2, r3 = 0). Therefore, the gradient measures G for all
three directions (three scalars expressed in a vector form) are:
G = [Gr1
, Gr2
, Gr3
]T = |∇rTp|centroid
= |(dNdr
)centroid
· Tp| (4.12)
where | | denotes the absolute values.
Stresses as the Gradient Measure
If using Gauss point quantities such as stresses, the first step is to acquire the
stress field in an element. And then the gradients of stress field at the centroid can be
evaluated. For example, for a hex8 element that has 2 × 2 × 2 Gauss points, the stress
field within the element is
91
S = 0.577350269189626 × N · S (4.13)
where N is the shape function, S = [S1,S2, ...,S8]T is the array of Gauss point
magnitudes. And
∇rS =dS
dr=
d(N · S)dr
=dNdr
· S (4.14)
where the local gradient ∇r is a 3 ∗ 1 vector.
Therefore, the gradient measures G for all three directions (three scalars expressed
in a vector form) are:
G = [Gr1
, Gr2
, Gr3
]T = |∇rS|centroid= |(dN
dr)centroid
· S| (4.15)
where | | denotes the absolute values.
92
4.4.3 Evaluation of refinement level
For each specific simulation, a desired permissible gradient Gp
is set at the start
of the analysis. The need to refine or coarsen each element is determined by comparing
the magnitudes of the gradients:
Ri= log2
Gri
Gp
(4.16)
where Ri(i = 1, 2, 3) determines the need to refine or coarsen in each direction. If
Ri
is positive, the element requests to be refined in the specific direction. On the other
hand, a negative number of Ri
suggests to coarsen the element.
Among the benefits brought by these derived equations are:
1. they can be easily derived for either orthogonal or skewed elements (Figure 2.4),
2. the computed gradient measures are completely unaffected by the difference in
global dimension sizes among all elements. Globally normalizing the element-wise
magnitudes can be avoided,
3. these independent element-wise calculation makes it practical to shorten the real-
time analysis length by utilizing multi-CPU parallelization if desired.
93
4.5 Moving Forced Refinement
In the AH-adaptive analysis, a starting mesh can be set to be the minimum
element densities whenever and wherever the Gauss points are sufficient to integrate
the system energy, as this also benefits an analysis by saving more CPU usage. So the
starting mesh may be very coarse. However, because
• for the time increments when the Gauss points in such a coarse initial mesh can
be too distant from the highest energy concentration to sufficiently integrate the
heat source,
• a forward re-meshing technique is utilized to significantly reduce the computational
cost compared to iterative re-meshing techniques.
moving forced refinement within spherical regions moving along with the heat
source(s) (see Figure 2.5) is introduced to trigger and enforce sufficient element densities.
The dual, or multiple if needed, moving spheres guarantee different degrees of refinement.
The concepts and utilization of the moving forced refinement in an AH-adaptive
mechanical (structural) analysis is the same as that of an AH-adaptive heat transfer
(thermal) analysis. The detailed illustrations can be seen in the previous work [35].
4.6 Element Coarsening
For both thermal analyses and mechanical analyses, the AH-adaptive FE scheme
processes element coarsening before refining. This sequence reduces the number of tem-
porary nodes and elements than the opposite way. An element is only allowed to coarsen
94
with its pair element if 1) the element has the gradient measure suggesting a larger size,
and 2) the suggested coarsening can not go below the necessary generation enforced by
the moving spheres.
4.6.1 Mutually Coarsenable Elements
In the AH-adaptive scheme, coarsening does not depend on the sequence of how
elements were refined. Therefore, how elements are coarsened does not need to be bound
with (sometimes can even be “hampered” by) the refining history for any element. Thus,
there can be more flexibility in re-meshing. However, coarsening without any reference
could induce an undesirable situation, such as that illustrated in Figure 2.8(a1). Element
A can not be coarsened with element B or C no matter how much coarser it desires,
though the elements are of exactly the same generations in all directions. In the example
elements B, C can still coarsen mutually, however, this clearly will not help coarsening
element A. Note that the elements could be coarsened into a single element whenever
they need to, if they are coarsened in the proper manner (Figure 2.8(a2)).
To make elements able to appropriately coarsen together with their “pair” (or
can be referred to as “sibling”) elements, regardless of what element generations they
are, three index numbers (I1, I2, I3) corresponding to the three local directions, are
assigned to every element and associated with current element generations (G1, G2, G3).
The previous paper on thermal analyses [35] illustrates how to use these information to
accurately determine the mutually coarsenable elements.
95
4.6.2 Transferring Data
After each coarsening, the original two elements will be deactivated. Therefore,
the data on the old elements need to be transferred. In an AH-adaptive mechanical
analysis, the primary categories of data to transfer for the finite element entities are:
1. Node-wise quantities: nodal solutions from the previous time increment, and bound-
ary conditions such as prescribed displacements.
2. Surface-wise quantities: boundary conditions such as surface pressure.
3. Element-wise quantities: Gauss point quantities, and remaining generations to
refine/coarsen, etc.
4.6.3 Processing the Nodes and Elements
Element coarsening never creates new nodes. Neither does it involve immediate
transfer of nodal quantities. Also note that a node can not be deactivated simply because
of element coarsening, as an adjacent element may still be using the node. Eliminating
any node can only be operated when the re-meshing procedure is complete, if not any
element is using it at all.
For a new generated element, the surface- and element-wise quantities (Section
4.6.2) are processed as:
a) Surface-wise quantities: Properties on the surfaces of a new element will inherit
from those of the corresponding old element surfaces.
b) Element-wise quantities:
96
1. Remaining generations to refine/coarsen:
if two elements A and B are to be coarsened in rn
direction, and
element A: (RA1, R
A2, RA3)
element B: (RB1, R
B2, RB3), where R
A′s and R
B′s are the remaining genera-
tions to re-mesh in all three directions for elements A and B, respectively, the
corresponding numbers for the new element C are
RCi
=
Int(R
Ai+R
Bi2 ) if i = 1 − 3 and i �= n
Int(R
Ai+R
Bi2 ) + 1 if i = n
(4.17)
where RC′s are how many generations element C needs to be re-meshed, i corre-
sponds to local directions, and Int() is taking the integer value.
2. Gauss points quantities: the magnitudes on a new Gauss point will be interpolated
from old Gauss points. Figure 4.4(a) shows an example for a specific element
coarsening.
4.6.4 Sequence of Element Coarsening
The AH-adaptive scheme processes all coarsenable elements in one specific local
direction (e.g. r1-direction) first. After this certain direction is finished, a second local
direction and then the third direction are operated sequentially. The reasons why the
sequence of directions matter, and how the starting direction is determined are illustrated
in [35].
97
r1
r2r3
r1-coarsening
XX
XX
XX
XX
##
##
##
##
: Gauss points on the new elemnt
: Gauss points on 1st old element
: Gauss points on 2nd old element
X
#
X
X
(a)
r1-refining
XX
: Gauss points on the new elemnts
: Gauss points onthe old elementX
X
X
(b)
XX
XX
XX
Fig. 4.4. Gauss point interpolations for (a) coarsening, (b) refining
98
4.7 Element Refining
4.7.1 Creating Entities
Anisotropic refinement creates two new elements, and the original element need
to be deactivated. For examples consisting of hex8 elements, a maximum of four new
nodes can be generated. However, if an existing node used by other element(s) has been
defined at the same coordinates, duplicating nodes is not allowed in the AH-adaptive
scheme.
4.7.2 Index and Generation Numbers
The index and generation numbers for the two new elements a and b are:
If refined in rn
direction, then
Iak
= Ibk
Gak
= Gbk
for k = 1, 2, 3, k �= n (4.18)
Ian
= 2 ∗ Iorigin,n
− 1
Ibn
= 2 ∗ Iorigin,n
(4.19)
Gan
= Gbn
= Gorigin,n
+ 1 (4.20)
4.7.3 Transferring Solution Field and Boundary Conditions
For refining an element,
1) Node-wise quantities: If a new node is generated, e.g. node 9 in the middle of
two original nodes 1 and 2,
99
ui,9 =
12(u
i,1 + ui,2)
where i = 1–3 represents x-, y-, and z-direction respectively.
2) Surface-wise quantities: Surfaces of the original element may either be split
onto the two refined elements or remain a whole piece, depending on the refinement
direction. Boundary conditions are transferred accordingly.
3) Element-wise quantities:
1. Remaining generations to refine/coarsen:
if an element is refined in rn
direction, with (R1, R2, R3), where R′s are how many
generations for this element to re-mesh, then
RAi
= RBi
=
Ri
if i = 1 − 3 and i �= n
Ri− 1 if i = n
(4.21)
where RA′s = R
B′s are the remaining generations of elements A and B to be
refined/coarsened.
2. Gauss points quantities: On the basis of old Gauss points, The example of Figure
4.4(b) illustrates that magnitudes on new Gauss points are calculated through
either interpolation or extrapolation. coarsening.
100
4.7.4 Sequence of Element Refining
The difference of resulted meshes between various sequence of operating directions
(in Section 4.6.4, or [35]) does not occur at element refinements, because each refining
only involves self-dividing regardless of its neighboring elements.
4.8 Identification of Dependent Nodes
If adjacent elements have different element generations, dependent (also referred
to as “constrained”) nodes take place. Illustrations for the causes, and how to accurately
handle these nodes in the system of equations with condensation and recovery method
[28], can be seen in [35]. Note that the difference between a thermal and a mechanical
analysis in the AH-adaptive scheme is the number of DOFs per node. In a structural
analysis, all DOFs on a dependent node have to be constrained as well.
4.9 Pre-processing for the Condensed System
4.9.1 Determination of Constraint Equations
As more details illustrated in [35], each dependent node can be regarded to be
constrained by the four corner nodes of the surface on which it locates. Hence the
constraint equation is in the form of:
ui,d
=(1 − r1)(1 − r2)
4ui,A
+(1 + r1)(1 − r2)
4ui,B
+(1 + r1)(1 + r2)
4ui,C
+(1 − r1)(1 + r2)
4ui,D
(4.22)
101
where r1 and r2 are the iso-parametric coordinates of the constrained node on
the surface, i = 1–3 correspond to the three spatial dimensions, ud’s is the solutions
(displacements) of the dependent node, uA
’s, uB
’s, uC
’s, uD
’s are those on the four
corner nodes which form that element surface.
And if a dependent node locates on an edge of an interface, Equation (4.22) will
reduce to two terms only:
ui,d
= αui,e1 + βu
i,e2 (4.23)
where α and β are the coefficients corresponding to two edge nodes e1 and e2
respectively, from Equation (4.22).
Condensing an original tangent matrix induces important effects which are intro-
duced and illustrated in the following sections.
4.9.2 Nonzero Fill-ins in Sparse Tangent Matrix for AH-Adaptive Mesh
The current research on the AH-adaptive analysis utilizes the IBM WSMP solver
[30], a sparse matrix solver which only utilizes information from the nonzero components
and saves it into linear arrays in order to reduce computational overhead. The nonzero
enforcement effect, which will be illustrated in this section, for the condensed matrix is
essential because it is necessary to determine the exact location of all nonzero positions
within the matrix in order to properly use the solver.
The condensation method ([28]) actually can be perceived as splitting the columns
and rows of the constrained DOFs into the columns and rows of the DOFs on which
102
they are dependent. To illustrate the perception, an example for the expanded number
of DOFs per node in a mechanical analysis is demonstrated in Figures 4.5, 4.6 and 4.7.
1. On the basis of the mesh in Figure 4.5, the DOFs on nodes 2 and 11 need to be
constrained. Due to the expanded number of DOFs in a 3-D mechanical analysis,
and the boundary conditions (prescribed displacements), a reference of each DOF
and the equation number is included in the figure.
2. The original nonzero components in the tangent matrix are presented in Figure
4.6. Splitting the columns and rows of the constrained DOFs in the matrix induces
nonzero enforcement effects into some positions which are originally zeros.
3. Figure 4.7 shows the condensed tangent matrix, and all the nonzero components
including those caused by the nonzero enforcement effect.
Note that because this example mesh contains only three elements and 16 nodes,
the non-zeros appear very dense in the tangent matrix (Figures 4.6 and 4.7). For practical
structures, they contain more elements and DOFs so that the non-zeros will be sparse.
4.9.3 Residual Array
The similarity and difference in condensing the residual array and tangent matrix
is illustrated in [35].
4.10 Gauss Point Quantities Balancing
New Gauss points, which replace all old Gauss points, will always be generated
during either element refinement or coarsening. However, 1) coarsening does not create
103
1
2
12
7
3
4
5
6
8
9
10
11
13
14
15
16
Node DOF Equation
1 x
y
zx
y
z
x
y
z
x
y
z
x
y
zx
y
z
x
y
z
x
y
z
x
y
z
x
y
z
x
y
z
x
y
z
x
y
zx
y
z
x
y
z
x
y
z
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
2
3
4
5
67
8
9
10
11
1213
14
1516
17
1819
20
2122
23
24
25
26
2728
29
3031
32
3334
35
36
3738
39
40
41x
yz
Fig. 4.5. Splitting the row and column of the dependent DOF.
104
: nonzero elements X
Original tangent matrix for the system of equations
Fig. 4.6. Nonzero fill-in effect induced by condensing the tangent matrix.
105
: original nonzeros in the stiffness matrix
: nonzero enforcements (fill-ins) due to the condensation effect
X
Condensed tangent matrix
Equation
Equatio
n
number i
n
number i
n
origin
al matri
x
origin
al mat
rixNew
equation n
umber
New equat
ion n
umber
Fig. 4.7. Nonzero fill-ins in the condensed matrix.
106
new nodes at all, and 2) refining does not necessarily involve with generating new nodes
(Sections 4.6 and 4.7). The nodal solution magnitudes with which to start for the next
time increment may be inconsistent with the Gauss point magnitudes.
Figure 4.8 shows an example of the possible inconsistency:
1. Figure 4.8(a) contains elements which are to be re-meshed. At this time increment,
one element is going to be refined to form a new adaptive mesh in Figure 4.8(b).
New Gauss points for the element are generated with data transferred. Meanwhile
all DOFS become free (unconstrained).
2. At a later time increment, two elements in Figure 4.8(b) are to be coarsened. Note
that new Gauss points of the coarsened elements are processed at the re-meshing
stage. And there are no new nodes created at all, all nodal solutions inherit from
those in the previous time increment.
3. However, after the new adaptive mesh is formed and upon identifying constrained
nodes, the mesh (Figure 4.8(c)) has two constrained nodes. The nodal solutions
on these constrained nodes now are forced to satisfy their individual constraint
equations, regardless of how much the nodal magnitudes have to be adjusted.
Therefore, on the elements which were just coarsened, the distributions of Gauss
point quantities (such as stresses, strains) become inconsistent with nodal quantity (dis-
placement) magnitudes. This inconsistency:
• does not occur in a thermal analysis, as there are no Gauss point quantities to
transfer during re-meshing. And the calculation of heat source loads for each
107
element on Gauss points is performed after the adaptive mesh information (such
as identifying constrained nodes) has been processed.
• may deteriorate convergence, especially in mechanical analyses of elasto-plasticity
or applying large deformation theory, which already have slower convergence rates
or are easier to diverge originally.
• may also give incorrect solutions, if solving the incremental equation system start-
ing with an incorrect combination of nodal and Gauss point quantity magnitudes.
To combat this, after a new adaptive mesh is determined and all constraint equa-
tions are acquired, the magnitudes of Gauss point quantities are adjusted. By using the
displacement results, and temperature inputs if a thermal-mechanical analysis such as
welding simulations, which 1) are from the previous time increment, and 2) satisfying
the constraint equations of the new adaptive mesh, the system of equations is solved
with one iteration starting with the interpolated Gauss point quantities. This yields a
new magnitude distribution for the Gauss points, balanced with the nodal quantities.
4.11 Recovering the Condensed DOFs
After the condensed system has been solved, the entire solutions for the origi-
nal/complete system can be acquired by applying the constraint equations to calculate
the magnitudes of the constrained DOFs.
108
(a)
(b)
(c)
: elements which are to be / have been refined
: elements which are to be / have been coarsened
: elements which remain unchanged
: new constrained nodes
Fig. 4.8. Balancing between nodal and Gauss point quantities
109
4.12 Convergence Efficiency Improving Strategy
Among the causes which make a mechanical analysis slower to converge are such
as plasticity, and large deformation simulation. This tendency may also induce diver-
gence at times during the entire simulation. (The algebraic equation system solving is
considered to be divergent also if not converging after a certain number of iterations.) In
ordinary finite element analyses, a time step is cut back upon diverging. Adaptive mesh
analysis process new mesh information (re-meshing, identifying constraint equations,
non-zero fill-ins determinations, etc.) each time a new mesh is generated. Therefore, a
cutback means spending more CPU usage on re-doing the mesh. As shown in Figure
4.1 of block group for “convergence efficiency improving”, to save the computational
cost, the AH-adaptive analysis scheme does not generate a new adaptive mesh if there
is divergence occurred. Instead, the scheme utilizes this adaptive mesh for the cutback
time step. The element density will be adequate for the cutback as long as the sizes of
moving forced refinement (Section 2.5) are large enough to include a tolerance for the
need of a few consecutive time steps.
Having constrained DOFs tends to make the equation system slower to converge.
This adds to the occurrence possibility that utilizing the same mesh for the cutback(s)
may not prevent the system from diverging. Therefore, to help a mechanical analysis
excess the occasional convergence hurdle, a maximum number of allowed cutbacks with
the same mesh in set-up for an AH-adaptive mechanical analysis. If the system still
diverges after the allowed number of cutbacks, the AH-adaptive scheme recalls a pre-
viously convergent mesh (node coordinates, solutions, connectivity, etc.) for this time
110
increment. Through this, the system is not locked in divergence and can step on until
the entire simulation is finished.
Thus, the AH-adaptive scheme also saves the information of the latest adaptive
mesh which has the system convergent, when this time increment is finished. This mesh
may be recalled if needed later on.
111
Chapter 5
Mechanical Analyses Numerical Examples
5.1 Linear Weld Path — Comparison of Static and AH-Adaptive Anal-
ysis
In order to 1) evaluate the computational cost using both the static and the
AH-adaptive schemes, and 2) verify that the adaptive analysis results match the static
analysis results, comparisons between the two analyses are performed.
The structural analysis is based on the temperature results from the same model
in [35] as inputs. Figure 5.1 depicts both the original mesh, and the boundary conditions
(prescribed displacements):
• At the front end, the node at the bottom of the mid point along y-direction is fixed
with δy = δz = 0.
• At the opposite end, the node at the bottom of the mid point is fixed at all DOFs.
And a neighboring node is fixed with δz = 0.
Material properties for ASTM 131 grade EH-36 steel are used [32]. Eight-node
hexagonal brick-type elements (hex8) are utilized in the analyses.
112
5.1.1 Hardware and Software
The simulations were performed on a SGI altix 350 system with 8 CPUs. The
software used in this study is an in-house finite element code written in Fortran 90.
An implicit solution scheme and the Newton-Raphson method were used to solve the
non-linear problems in an iterative fashion.
5.1.2 Analysis Results
Figure 5.2 shows the structural response simulation of the static mesh (Figure
5.1) analysis, using the temperature results presented in the previous paper [35]. In the
figure, the blue shaded elements are the deformed shape at t = 3600 sec, in reference
with the undeformed shape. An adaptive mesh analysis is also performed. Figure 5.3
depicts the initial coarse mesh for the structural analysis. The deformation results at t
= 3600 sec is shown in Figure 5.4, with red shaded elements of the deformed in reference
with the undeformed shape.
5.1.3 Comparison
Table 5.1 gives the computation statistics for both the static and the AH-adaptive
analyses. Note that the presented examples demonstrate the ability to reduce the analysis
CPU time by 1131.25 % compared to the conventional static solution.
113
X
Y
Z
23
X
Y
Z
Opposite end
Front end
Fig. 5.1. Static mesh with boundary conditions
114
X
Y
Z
1.71+00
default_Deformation :Max 1.71+00 @Nd 122
X
Y
Z
Fig. 5.2. The deformation results (mm) of the static mesh analysis. Magnificationfactor = 5.0.
115
X
Y
Z
X
Y
Z
Fig. 5.3. Initial mesh for the AH-adaptive analysis
116
X
Y
Z
default_Deformation :Max 1.25+00 @ Nd 25
X
Y
Z
Fig. 5.4. The deformation results (mm) of the adaptive mesh analysis. Magnificationfactor = 5.0.
117
Table 5.1. Comparison between the static and the AH-adaptive analyses on the modelStatic Mesh Analysis AH-Adaptive Analysis
Number of Nodes Statically 77979 Initial 240 Peak 28030Number of Elements Statically 70800 Initial 120 Peak 19174Total Number ofTime Increments
830 915
Total number if Iter-ations
5322 6578
Maximum displace-ment (mm)
1.71 1.25
Total Analysis CPUTime
105682 sec 72631 sec
Processing adaptivemesh information
(N/A) 52008 sec
Residual and tan-gent matrix assem-bling
10935 sec 4507 sec
Algebraic equationsolving
87310 sec 13478 sec
118
5.2 Combined Weld Path (Curved and Linear) — Evaluation of AH-
Adaptive Analysis Scheme
The AH-adaptive FE scheme is also applied to simulate a welding procedure on a
3ft× 3ft plate shown in Figure 5.5. Note also the heat source does not merely move in
linear paths, but also moves along a one-quarter arc of a circular path. In this case, the
heat source is a hybrid of laser-GMAW weld, with heat input parameters can be seen in
[35]. The material properties, hardware and software are the same as those described in
Section 5.1.
The performed structural response simulations are based on the temperature re-
sults acquired from the same structure of [35] — the example in Section “Combined
Weld Path (Curved and Linear)”. The deformation results are shown in the following
section.
5.2.1 Analysis Results
The experimental result of deformations is shown in Figure 5.6. The plate buckles
after applying the heat source. And the dominant buckling in this specific experimental
case is the third mode (the first three buckling modes depicted in Figure 5.7). Simulation
results at t = 3600 sec applying Total Lagrangian formulation for large deformation
analyses with different parameter settings are shown in Figures 5.8 and 5.9. Figure 5.8
uses the permissible gradient = 58, while Figure 5.9 sets to 400. Both analyses allow
elements to be refined to minima of 0.3mm along thickness direction, and 1.0mm for
the remaining two directions. As the figures show, buckling does also occur in both
119
simulations. Figure 5.8 captures the first buckling mode. Meanwhile the buckling mode
on Figure 5.9 is the second mode.
1. In experiments, imperfections of structure shape can induce different deformation
magnitudes, especially when buckling occurs.
2. In simulations of adaptive analyses, different mesh densities controlled by user
settings may also result in different buckling modes between simulations, because
of the different connectivities and therefore the allowed deformation shape through
the entire analyses.
3. Experimental boundary conditions usually will not be exactly identical to those
applied in a simulation. Thus, if the structure buckles, the maximum buckling
displacement magnitudes may differ much between experiment and simulation, or
even between simulations especially if they capture different buckling modes.
Solving this problem using a conventional static mesh is intractable due to:
• the large number of elements and DOFs which would be required to adequately
describe the structure.
To generate a static mesh for the 3ft×3ft plate (of the same size shown in Figure
5.5), the estimated node number is calculated from the static mesh of Figure 5.1
used in Section 5.1. The small structure in Figure 5.1 is cut from the 3 × 3 plate,
and contains only 152.4mm of the linear weld path portion. Its width is also only
304.8mm (less than the width 914.4mm of the entire plate). The total weld length
on the 3 × 3 plate is 609.6mm (two linear segments) + 215.8mm (circular path)
120
X
Y
Z X
Y
ZX
Y
Z X
Y
Z
Fixed at the bottom, x=y=z=0
Fixed at the bottom, x=0Fixed at the bottom, z=0
Fixed at the bottom,y=0
Fig. 5.5. The plate and the initial mesh.
121
Fig. 5.6. Experimental buckling results.
122
Mode 3
Mode 1Mode 2
1
2
3
Pure Angular Distortion
Fig. 5.7. 1st – 3rd buckling modes and pure angular distortion.
123
X
Y
Z
8.49+00
default_Deformation :Max 8.49+00 @Nd 25
X
Y
Z
Fig. 5.8. The deformation result (mm) at t = 3600 sec, with permissible gradient (peaktemperature) = 58 ◦ C. (Magnification factor = 2.5)
124
X
Y
Z
1.35+01
default_Deformation :Max 1.35+01 @Nd 21
X
Y
Z
Fig. 5.9. The deformation result (mm) at t = 3600 sec, with permissible gradient (peaktemperature) = 400 ◦ C. (Magnification factor = 2.5)
125
= 825.4mm. Therefore, considering there are 77979 nodes on the mesh of Figure
5.1, using similar mesh densities for the 3ft × 3ft plate would give the estimated
node number to be “above” 77979/152.4 ∗ 825.4 = 422335.
• the complexity of making transition elements (between the high element density
areas and elsewhere), especially because of the existence of the circular weld path.
The CPU usages and other statistics using different sets of parameters (such as
the sizes of the moving forced refinements) with the AH-adaptive scheme is presented in
Table 5.2. Using a standard direct sparse solver, the computational cost to factorize and
solve the equation system is known to grow as O(n2) for 3D simulations (and O(n32 ) for
2D simulations) [34], where n is the total number of equations in the system. However,
benefitting from the huge reduction of DOFs, in the AH-adaptive analysis the equation
solving is not dominating the total computation. And the adaptivity overhead induced
by processing the adaptive mesh information is designed to be linear (O(n)). If the
necessary DOFs for a static mesh analysis is larger, the efficiency is expected to be
even much higher because the computational expense spent on decomposing the tangent
matrices will be reduced more significantly.
126
Table 5.2. Statistics of the AH-adaptive analysisCase number L1 L2Radii of EnforcedRefinement
11.987mm and 13.91mm 11.987mm and 13.91mm
Minimum Length ofElements
0.3mm in thickness 0.3mm in thickness
0.6mm elsewhere 0.6mm elsewherePermissible Gradient(Peak Temperature ◦C)
400.0 58.0
Total Number ofTime Increments
480 507
Number of Nodes Initial 440 Peak 22151 Initial 440 Peak 24084Number of Elements Initial 288 Peak 17983 Initial 288 Peak 16398Average number ofnodes per iteration
14152 15008
Total Analysis CPUTime
75107 sec 85731 sec
Residual and Tan-gent Matrix Assem-bling
50031 sec 57532 sec
Algebraic EquationSolving
14048 sec 16130 sec
Adaptive Mesh In-formation Processing
10168 sec 11094 sec
127
Chapter 6
Comparisons between using Condensation Theory,
Lagrange Multiplier and Penalty Method
for Constrained DOFs
Upon applying h-refinement strategies [20] in finite element simulations, depen-
dent (also referred to as “constrained”) nodes occur wherever adjacent elements have
different element generations. Setting the element generations to be identical (for exam-
ple, 0) for all elements on the starting mesh in Figure 2.11(A), Figure 2.11(B) shows an
example mesh containing constrained nodes due to the generation differences. The de-
pendent nodes need to satisfy the interpolation (linear for hex8 elements, and quadratic
for hex20 elements) on the interface shared with the adjacent elements. Thus, the DOFs
on these nodes must be constrained, and the original system of equations therefore has to
be adjusted before being solved to satisfy these constraints. A few methods are available
to account for the constrained DOFs:
• Condensation and recovery theory [28]
• Lagrange multiplier [38]
• Penalty Method [38]
The examples given in this paper use hex8 elements whenever a specific mesh is
needed. However, the utilization of these methods for dealing with constrained equations
apply to all element types.
128
6.1 Objective
The objective of this chapter is to compare the above methods which serve to
process an algebraic system of constrained equations. The comparisons are effective
not only when applying the AH-adaptive scheme [35, 39] for simulations that involve
multi-scale analyses, but also for other equation systems induced by h-refinement.
6.2 Determination of Constraint Equations
In Figure 2.12 consisting of hex8 elements, nodes 1–10 are constrained by the
corner nodes A–D on the surface of element a, because only linear interpolations will
be allowed on that surface. By using the corresponding iso-parametric coordinates (r1
and r2), every constrained node may be treated as depending on the four corner nodes
of the surface. Hence the constraint equation for a dependent node is:
Td
=(1 − r1)(1 − r2)
4TA
+(1 + r1)(1 − r2)
4TB
+(1 + r1)(1 + r2)
4TC
+(1 − r1)(1 + r2)
4TD
(6.1)
where r1 and r2 are the iso-parametric coordinates of the constrained node on the surface.
Td
is the temperature of the dependent node. TA
, TB
, TC
, TD
are the temperatures on
the four corner nodes forming the element surface.
For a dependent node located on an edge of an interface, Equation (6.1) will reduce
to two terms only, which correspond to the fact that this node is actually constrained by
the two end points of the edge. For nodes on these edge interfaces, the equations are:
129
Td
= αTe1 + βT
e2 (6.2)
where node d depends on two edge nodes e1 and e2, α and β are the coefficients corre-
sponding to the two edge nodes e1 and e2, respectively, from Equation (6.1).
For instance, in Figure 2.12 nodes 1 and 2 only depend on nodes A and B, while
nodes C and D do not affect them at all. Therefore,
T1 = 0.5 ∗ TA
+ 0.5 ∗ TB
(+0 ∗ TC
+ 0 ∗ TD
)
T2 = 0.25 ∗ TA
+ 0.75 ∗ TB
(+0 ∗ TC
+ 0 ∗ TD
)
6.3 Condensation and Recovery Theory
To compute a system containing some constraint equation(s), the condensation
(and recovery) theory takes the constrained variables (DOFs) out, in order to form a
equation system that has only free variables (DOFs). The condensed variables are recov-
ered by the constraint equations after the solutions of the free variables are calculated.
6.3.1 System Condensing
For a non-linear system, the incremental nodal solution is computed from the
algebraic system
Aδu = b (6.3)
130
where A is the tangent matrix with b being the negative of the residual, and δu is the
incremental solution during an iteration.
For each iteration, the system (Equation (6.3)) is processed by partitioning the
DOFs into
{δu} =
δur
δuc
(6.4)
where the subscript r stands for “retained”, and c stands for “condensed (out)”. Thus,
δur represents the actual DOFs to be retained, and δuc represents the condensed DOFs
of the dependent nodes. Thus, the entire partitioned non-linear system can be repre-
sented as:
Arr Arc
Acr Acc
δur
δuc
=
br
bc
(6.5)
The general representation of the constraint equations is given by
[Cr Cc
]
δur
δuc
={
Q
}(6.6)
where Cr
and Cc
are the coefficients for the retained and condensed nodes, respectively,
131
and Q is the constant in the system of constrained equations. For the AH-adaptive
analysis scheme, a constraint equation must have the form of
uc
=N∑
k=1C
kuk
(6.7)
where uc
is the solution of the constrained node. N is the number of nodes on which it
depends, and Ck
is the corresponding constraint coefficient.
The RHS terms in Equation (6.7) can be moved to the LHS of the equation:
uc−
N∑k=1
Ckuk
= 0 (6.8)
So that the constant term Q in Equation (6.6) must be zero.
Utilizing the equations representing the constraints, we now have:
[Cr Cc
]
δur
δuc
={
0
}(6.9)
So,
{δuc
}= −[ Cc
]−1[ Cr]{
δur
}= [ Crc
]{
δur
}(6.10)
where Crc = −C−1
cCr is the combined coefficient matrix.
132
By substituting Equation (6.10) into Equation (6.5),
[Arr + ArcCrc + CT
rcAcr + CT
rcAccCrc
] {δur
}=
{br + CT
rcbc
}(6.11)
the solutions for the “retained” DOFs{
δur
}is:
{δur
}=
[Arr + ArcCrc + CT
rcAcr + CT
rcAccCrc
]−1 {br + CT
rcbc
}(6.12)
And the “condensed” DOFs can then be recovered by Equation (6.10).
6.3.2 Example
In Figure 2.11, with 24 independent nodes and 4 constrained nodes, the constraint
equations for the dependent nodes are:
ua
=12(u1 + u2) (6.13)
ub
=12(u2 + u3) (6.14)
133
uc
=12(u4 + u5) (6.15)
ud
=12(u5 + u6) (6.16)
Equation (6.9) for this example system becomes
1 0 0 0 −12 −1
2 0 0 0 0 0 ... 0
0 1 0 0 0 −12 −1
2 0 0 0 0 ... 0
0 0 1 0 0 0 0 −12 −1
2 0 0 ... 0
0 0 0 1 0 0 0 0 −12 −1
2 0 ... 0
4∗28
ua
ub
uc
ud
u1
u2
u3
u4
u5
u6
u7
...
u24
28∗1
=
0
0
0
0
4∗1
(6.17)
134
Thus,
[Cc
]=
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
= [ I ]4∗4 (6.18)
[Crc
]= −[ Cr
] =
12
12 0 0 0 0 0 ... 0
0 12
12 0 0 0 0 ... 0
0 0 0 12
12 0 0 ... 0
0 0 0 0 12
12 0 ... 0
4∗24
(6.19)
Therefore, the retained, and the condensed, DOFs can be solved using Equation
(6.12).
6.3.3 Restrictions
To utilize the condensation (and recovery) theory in AH-adaptive analyses, the
followings need to be taken care of:
135
1. the tangent matrix is condensed to a smaller size [28, 35], so allocation for the
matrix of the corresponding dimension is necessary.
2. in practical simulations, the tangent matrix is generally a sparse matrix so that
a linear sparse matrix solver can be used to enhance computational efficiency.
However, condensing the tangent matrix induces the effect of nonzero fill-in which
is illustrated in Tsau et al. [35]. Special considerations have to be performed to
accurately solve the system.
6.4 Lagrange Multiplier
Lagrange multiplier method is a widely used procedure for imposing constraints
onto a system. The method adds the constraints, and operates on, the original variational
formulation of the system, in order to acquire the adjusted equations to solve.
6.4.1 Equation Derivation
Consider the variational formulation of a discrete structural model,
Π =12uTAu − uTb (6.20)
and
∂Π∂u
i
= 0 for all i (6.21)
136
where Π is the total potential (also referred to as “functional”) of the system, u is the
(incremental) solutions of all DOFs, ui
is the (incremental) solution of the i − th DOF,
A is the tangent matrix, and b is the residual.
Assume that there are m linearly independent constraints Cu = Q to be imposed,
where C is a coefficient matrix of order m×n, n is the total number of equations in the
original system, Q is an m × 1 array of constants. By applying the Lagrange multiplier
method, the variational formulation of a discrete structural model [40] is modified as
Π∗(u, λ) =12uTAu − uTb + λ
T(Cu − Q) (6.22)
where λ is a vector of m Lagrange multipliers.
Letting δΠ∗ = 0, and because δu and δλ are arbitrary,
A CT
C 0
u
λ
=
b
Q
(6.23)
As illustrated in Equations (6.7) and 6.8 (or Equation (6.17) from a numerical
example), in the constraint equations it must be Q = 0.
And,
A CT
C 0
u
λ
=
b
0
(6.24)
Multiplying −CA−1 with the first row, and adding to the second row,
137
A CT
0 −CA−1CT
u
λ
=
b
−CA−1b
(6.25)
The vector λ is acquired first by
λ = (−CA−1CT)−1(−CA−1b) (6.26)
This allows to determine u:
u = A−1(b − CTλ) (6.27)
6.4.2 Restrictions
The Lagrange multiplier method induces many drawbacks:
1. though the construct of A is generally a sparse matrix, −CA−1CT in Equation
(6.26) is a dense matrix, so that it can not be operated by a sparse matrix solver.
Figures 6.1 shows a mathematical example of how −CA−1CT forms a dense/full
matrix, even if A−1 is set to be sparse.
Note that for using a standard direct sparse solver, the computational cost to fac-
torize and solve the equation system is known to grow as O(n2) for 3D simulations
[34], where n is the total number of equations in the system. However, the opera-
tions on the dense matrix −CA−1CT is of order O(m3), where m is the number
of constrained equations.
138
2. as Equations (6.26) and (6.27) show, the Lagrange multiplier method requires
computations of more matrix inverses (in addition to other arithmatic operations)
which are the major contributing factor to computational cost especially for large
structures,
3. the tangent (stiffness) matrix A also retains the total number of DOFs, and does
not benefit from the reduction in dimension realized because of the constrained
DOFs in applying condensation theory.
6.5 Penalty Method
Similar to the Lagrange multiplier method, penalty method combines the imposed
constraints to the original variational formulation of the system. However, instead of the
additional variable λ, the penalty method introduces a constant penalty number α to
derive for an modified equation system to solve.
6.5.1 Utilizing Penalty Number in a System
Starting also from Equations (6.20) and (6.21), the variational formulation of a
discrete structural model, and the assumption that there are m linearly independent
constraints Cu = Q to impose. In the penalty method, the variational formulation of a
discrete structural model [40] is formulated as follows: