Robust and efficient nonlinear structural analysis using the central difference time integration scheme

Robust and efficient nonlinear structural analysis
using the central difference time integration scheme
Reagan Chandramohan
Jack Baker
Greg Deierlein
19 June 2017
1st European OpenSees Days
Porto, Portugal

Background and Motivation
T
Sa(T)
Conditional mean spectrum
In nonlinear structural analysis, emphasis
is typically on formulating the problem
correctly
Creating a detailed and accurate
structural model
Selecting an appropriate number of
hazard-consistent ground motions
Reagan Chandramohan Central difference scheme University of Canterbury 2

x
f (x)
x0
x1
x
f (x)
x0
x1
Traditionally, lesser attention is paid to the
process of solving the formulated problem
correctly
Handling numerical non-convergence
Treatment of ill-conditioned matrices
Precision of computations
Collapse detection criteria
Differences between analysis software
Analysis efficiency, similarly, receives
relatively little attention

Nonlinear structural analysis is most frequently conducted using implicit time
integration schemes, like the Newmark average acceleration scheme
Implicit schemes are iterative in nature and often fail to converge, especially
when analysing large, complex models under intense, long duration ground
motions
Attempts to force convergence are computationally expensive and not always
successful
These issues impede progress towards being able to routinely analyse
multiple realisations of complex structural models using large numbers of
ground motions by effectively harnessing high-performance computers

Objectives
Explore the use of the explicit central difference time integration scheme as an
alternative to implicit schemes like the Newmark average acceleration scheme
Develop parallel algorithms to efficiently conduct computationally intensive
analyses like multiple stripe analysis (MSA) and incremental dynamic
analysis (IDA) on high-performance computers

Comparison of time integration schemes
Newmark average acceleration Central difference
Implicit scheme
4
∆t2
M +
2
∆t
C ui+1 =
p[M, C, ∆t,ui, ˙ui, üi, (üg)i] − fi+1
Solves for equilibrium at end of time step
Requires solution by iteration; convergence is
not guaranteed
If convergence fails
try other solution algorithms, e.g.
Modified Newton-Raphson,
Newton-Raphson with initial stiffness
try other implicit schemes with
algorithmic damping
try reducing ∆t
These attempts are time-consuming
If they all fail, structural collapse is declared
even if collapse deformation threshold is not
exceeded
Explicit scheme
1
∆t2
M +
1
2∆t
C ui+1 =
p[M, C, ∆t,ui, ui−1, (üg)i] − fi
Solves for equilibrium at beginning of time step
No iteration required
If C is constant (and diagonal), matrix needs to
be factorised only once
Amenable to parallelisation by domain
decomposition

Comparison of time integration schemes
Newmark average acceleration Central difference
Unconditionally stable
∆t limited by accuracy, not stability
Can use relatively large ∆t
(∼ 10−3 s to 10−2 s), which is usually reduced
upon encountering non-convergence
Not easy to predict duration of analysis
Conditionally stable
∆t ≤ Tmin/π for stability
∆t used is usually relatively small
(∼ 10−4 s)
Tmin is usually unchanged in inelastic range
Mass/moment of inertia must be assigned to all
degrees of freedom
Impractical to use rigid elements or penalty
constraints
Easy to predict duration of analysis
(enables static parallel load balancing)

Structural model
Column hinge
Beam RBS hinge
Joint panel
Leaning
column
9-story steel moment frame
building from SAC steel project
2d concentrated plastic hinge
model created in OpenSees
Plastic hinges follow
Ibarra-Medina-Krawinkler
bilinear hysteretic model
Fundamental elastic modal
period is 3.0 s
Conducted IDA separately using
Newmark average acceleration
and central difference schemes

Incremental dynamic analysis (IDA)
0.00 0.02 0.04 0.06 0.08 0.10
Peak story drift ratio
0.0
0.2
0.4
0.6
0.8
1.0Sa(3.0s)(g)
IDA curves
0.0 0.5 1.0
Probability of collapse
Collapse fragility
curve
Empirical CDF
Used FEMA P695 far field record set,
containing 44 ground motions
Each ground motion is progressively scaled
to higher intensity levels until it causes
structural collapse
Collapse fragility curve is fit to the
distribution of ground motion intensity levels
at which structural collapse occurs

IDA curves bifurcate due to non-convergence
Difference in estimated collapse capacity > 10 % for 12 out of 44 ground motions
0.00 0.02 0.04 0.06 0.08 0.10
0.00
0.25
0.50
0.75
1.00
Sa(3.0s)(g)
23 % lower
Central difference
Newmark avg. accel.
0.00 0.02 0.04 0.06 0.08 0.10
0.00
0.25
0.50
0.75
1.00
Sa(3.0s)(g)
32 % lower
Central difference
Newmark avg. accel.
0.00 0.02 0.04 0.06 0.08 0.10
0.00
0.25
0.50
0.75
1.00
Sa(3.0s)(g)
41 % lower
Central difference
Newmark avg. accel.
0.00 0.02 0.04 0.06 0.08 0.10
0.00
0.25
0.50
0.75
1.00
Sa(3.0s)(g)
34 % lower
Central difference
Newmark avg. accel.

IDA curves are similar when analyses converge
Difference in estimated collapse capacity < 1 % for 29 out of 44 ground motions
0.00 0.02 0.04 0.06 0.08 0.10
0.00
0.25
0.50
0.75
1.00
Sa(3.0s)(g)
Central difference
Newmark avg. accel.
0.00 0.02 0.04 0.06 0.08 0.10
0.00
0.25
0.50
0.75
1.00
Sa(3.0s)(g)
Central difference
Newmark avg. accel.
0.00 0.02 0.04 0.06 0.08 0.10
0.00
0.25
0.50
0.75
1.00
Sa(3.0s)(g)
Central difference
Newmark avg. accel.
0.00 0.02 0.04 0.06 0.08 0.10
0.00
0.25
0.50
0.75
1.00
Sa(3.0s)(g)
Central difference
Newmark avg. accel.

Effect on estimated collapse fragility curves
0.1 1
Sa(3.0 s) (g)
0.0
0.2
0.4
0.6
0.8
1.0
Probabilityofcollapse
Central difference
(µ =0.38 g)
Newmark avg. accel.
(µ =0.34 g)
Median collapse capacity is under-estimated by 10 % using the Newmark average
acceleration scheme
Similar effect expected on collapse fragility curves estimated using multiple stripe
analysis (MSA) as well

Comparison of analysis runtimes
One analysis using one ground motion
Time integration
scheme
Rayleigh
damping matrix
Type of
solver
∆t (s)
Analysis
runtime (min)
Newmark avg. accel.
low scale factor
w/o convergence attempts
αM + βKcurrent Sparse 50 × 10−4
1.0
Newmark avg. accel.
high scale factor
w/ convergence attempts
αM + βKcurrent Sparse ≤ 50 × 10−4
20.9
Central difference αM + βKcurrent Sparse 1.5 × 10−4
15.9
Central difference αM + βKinitial
Sparse
(factor once) 1.5 × 10−4
3.3
Central difference αM
Diagonal
2.9
Using Kinitial instead of Kcurrent in the Rayleigh damping matrix has been shown to
produce spurious damping forces
Other option is to use a modal damping matrix, which is also constant

Comparison of analysis runtimes
Entire IDA (using 160 processors and dynamic load balancing)
Time integration
scheme
Rayleigh
damping matrix
Type of
solver
∆t (s)
IDA runtime
(min)
Newmark avg. accel. αM + βKcurrent Sparse ≤ 50 × 10−4
118
Central difference αM + βKcurrent Sparse 1.5 × 10−4
154
Central difference αM + βKinitial
Sparse
32
Central difference αM
Diagonal
27
Only 1 out of 632 total analyses conducted using the Newmark average acceleration
scheme completed without any convergence errors
567 out of the 632 analyses completed using solution algorithms other than full
Newton-Raphson
23 out of the 632 analyses completed after reducing ∆t

Parallel algorithms
Developed parallel algorithms to conduct MSA and IDA on multi-core
computers and distributed parallel clusters
Dynamic load-balancing using a master-slave approach
Checkpoint-restart functionality
Implemented using features currently available in OpenSeesMP
(available to download from bitbucket.org/reaganc/msa_ida_parallel)
P0
Master process
P1 P2 P3 P4 P5 ... Pn
Slave processes
• ground motion
information
• abort/continue
command
• terminate
command
• collapse indicator
• status inquiry

Runtime and parallel efficiency of IDA algorithm
1 10 100 400
Number of processors, n
0.2
1
10
100
Runtime,T(n)(h)
No load balancing
Algorithm
1 10 100 400
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Parallelefficiency,E(n)
No load balancing
Algorithm
No load balancing: Records distributed to processors
Algorithm: Individual analyses distributed to slave processes

Runtime and parallel efficiency of IDA algorithm
1 10 100 400
0.2
1
10
100
Runtime,T(n)(h)
No load balancing
Algorithm
Algorithm
(oversubscribed)
1 10 100 400
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Parallelefficiency,E(n)
No load balancing
Algorithm
Algorithm (oversubscribed)
No load balancing: Records distributed to processors
Algorithm: Individual analyses distributed to slave processes
Algorithm (oversubscribed): As many slave processes as available cores

Conclusion
The explicit central difference time integration scheme is a robust and
efficient alternative to commonly used implicit time integration schemes like
Newmark average acceleration
Robust: immune to numerical non-convergence
Efficient: shorter runtimes despite using a smaller ∆t
Most efficient when using constant (and diagonal) C matrix
More amenable to parallelisation
Explicit schemes are preferred in blast and crash simulations involving large
nonlinear deformations; such nonlinear structural analyses should also benefit
Disadvantages
Mass/moment of inertia must be assigned to all degrees of freedom
Impractical to use rigid members or penalty constraints
Developed efficient parallel algorithms to conduct MSA and IDA on parallel
computers using OpenSeesMP

Thank you!
Reagan Chandramohan
Lecturer
University of Canterbury
Christchurch, New Zealand
reagan.c@canterbury.ac.nz
reaganc.bitbucket.io

Robust and efficient nonlinear structural analysis using the central difference time integration scheme

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Robust and efficient nonlinear structural analysis using the central difference time integration scheme

Similar to Robust and efficient nonlinear structural analysis using the central difference time integration scheme (20)

More from openseesdays

More from openseesdays (20)

Recently uploaded

Recently uploaded (20)

Robust and efficient nonlinear structural analysis using the central difference time integration scheme