Parallel Processing Techniques for the Nonlinear Dynamic Analysis of Structures

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Parallel Processing Techniques for the Nonlinear
Dynamic Analysis of Structures
Elisa D. Sotelino
School of Civil Engineering Purdue University
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Statement of the Problem
u(t)
fext(t)
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Newmark Method (1959)
Finite difference formulas
Unconditional Stability
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Newmark Family of Methods
 
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General Solution Scheme

• Estimate displacements, velocities and
  accelerations using Newmarks finite difference
  approximations

• Compute the displacement increment as

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General Solution Scheme (Cont.)

• Update the displacements, velocities and
  accelerations using the obtained corrections as

• Increment time nn1, tth and go to the second
  step if not converged

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Central Difference Method
Time-step size for stability
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Parallel Solution Algorithms

• Substructuring-Based Methods
• Domain-Splitting Methods
• Parallel Central Difference Method
• FETI Algorithm
• GI Algorithm
• MIGI Algorithm

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Domain Partitioning

• Automatic Domain Partitioning
• Usually based on Graph Theory
• Important especially for large 3-D meshes
• Manual Domain Partitioning
• Fine tuning partitioning

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Substructuring-Based Methods

• Divide the structure into substructures
• Condense out interior DOF (in parallel)
• Solve the reduced system for the interface DOF
  (in parallel)
• Recover interior DOF solution from the interface
  DOF

However communication is necessary since the
reduced stiffness matrix and load vector have
overlapping coefficients
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Domain-Splitting Methods
Parallel Central Difference Scheme (Hajjar
Abel 1989)
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Finite Element Tearing and Interconnecting (FETI)
Algorithm (Farhat Roux 1991)
is the subdomain connectivity matrix ? are
the Lagrange multipliers s 1, 2, , Ns
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Performance of the FETI Algorithm
3-D Example
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Group Implicit (GI) Algorithm (Sotelino et al.
1990)
Partition the mesh into
subdomains


1.
For each
subdomain

Solve using an implicit scheme and
accept solution for the interior
DOFs
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GI Algorithm (Cont.)
2.
Enforce compatibility of interface DOFS
3.
Accept solution
assemble displacement vector
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GI Algorithm - Characteristics

• Same stability characteristics as underlying
  implicit procedure

Consistent
mass averaging rule

For accuracy time
-
step size is limited by

a Courant type condition
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GI Algorithm - Characteristics
Strengths

Highly modular
l
ow communication overhead
Weakness

Deterioration of accuracy (unreliable for
practical h)
Reason for inaccuracies

Mass averaging rule introduces unknown amount
of interface reactive forces
equilibrium is not satisfied
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Modified Iterative Group Implicit (MIGI)
Algorithm (Modak Sotelino 1997, Dere
Sotelino 2002)
Partition the mesh into subdomains

• For each time-step

• Solve each subdomain estimate interface
  reactive forces.

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MIGI Algorithm (Cont.)
2. Iterate until convergence
2.1 Solve each subdomain for the computed
corrective forces
2.2 Enforce compatibility
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MIGI Algorithm (Cont.)
2.3 Re-evaluate interface reactive forces from
residual interface displacements
2.4 Assemble interface reactive forces
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MIGI Algorithm (Cont.)
2.5 IF THEN convergence is achieved ELSE
distribute interface residual forces among
DOFs
2.6 Update interface force vector
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Nonlinear MIGI Algorithm

• Two possible schemes
• Combined iterations nonlinear and
• MIGI iterations are combined
• Separate iterations nonlinear and MIGI
• iterations are isolated

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Iterative Integration Schemes
Separate Scheme
Sequential Solution
Load
Rn1
Rn
Combined Scheme
Displac.
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Combined Scheme
Separate Scheme
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Performance Comparison
Mass density 2.26e-2 lb-sec2/in Cross-section
W12x14 standard section
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Accuracy Verification
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Nonlinear Iterations
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Total Analysis Time
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Total time comparison
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Average times per main iteration
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Total time comparison
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Average times per main iteration
Note Total time (sequential) 188 sec
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Demonstration
Nonlinear Transient Analysis of a 20-story SAC
Building

• 266 Nodes, 391 Elements
• Fiber beam-column elements
• SAC building model data Gupta,
• A. and Krawinkler, H. (1999)
• Northridge Newhall Earthquake
• Lumped mass, Rayleigh damping
• Material and Geometric
• nonlinearity
• IBM SP multicomputer

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Domain partitioning (MPE)
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Top Story Displacement
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Speed-up
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Asymptotic iteration performance
Time step size Dt 0.01 sec
Number of Processors
Total analysis time (sec)
CPU time/Dt (sec)
Number of nonlinear iterations
1 2 3 4 5 6
15,031 8,659 5,762 4,414 3,956 3,458
7.50 4.32 2.87 2.20 1.96 1.72
4,323 4,321 4,321 4,312 4,315 4,307
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Demonstration 3D
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Domain Partitioning (MPE)
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Top Story Displacement
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Speed-up
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Future Research Directions

• Improved Automatic Domain Decomposition
  Algorithms
• Dynamic Load Balancing
• New algorithms for Grid Computing (communication
  is major handicap)