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Numerical Simulation of 3D Fully Nonlinear Waters Waves on Parallel Computers

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Numerical Simulation of3D Fully Nonlinear Waters Waveson Parallel Computers

Xing Cai

University of Oslo

- Mathematical model
- Numerical scheme (sequential)
- Parallelization strategy (domain decomposition)
- Object-oriented implementation
- Numerical experiment

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- Fully nonlinear 3D water waves
- Primary unknowns:

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- Physical domain:
- Transformation: (a fixed domain)

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- Operator splitting
- At each time level:
- FDM for updating free surface conditions
- FEM solution of an elliptic boundary value problem in

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- Elliptic boundary value problem - most CPU intensive
- Resulting system of linear equations
- Preconditiong

Computational cost

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N- number of unknowns

Starting point: an o-o water wave simulator

(built in Diffpack: C++ environment for scientific computing)

How to do the parallelization?

- Different approaches on different levels:
- Automatic parallelization
- Parallelization on the low matrix-vector level
- Parallelization on the level of simulators

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- Domain Decomposition
- Divide and conquer
- Solution of the original large problem through iteratively solving many smaller subproblems --solution method or preconditioner
- Flexible -- localized treatment of irregular geometries, singularities etc
- Very efficient numerical methods -- even on sequential computers
- Suitable for coarse grained parallelization

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- Alternating Schwarz method for two subdomains
- Example: solving an elliptic boundary value problem
- in
- A sequence of approximations
- where

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- Additive Schwarz Method
- Subproblems are of the same form as the original large problem, with possibly different boundary conditions on artificial boundaries.
- Subproblems can be solved in parallel.

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Example:

Solving the Poisson

problem on the unit

square

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- Coarse Grid Correction
- Important for good DD convergence
- Run on each processor, shared with subdomain simulators on the same processor

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- Parallel Computing
- efficiency relies on the parallelization
- Domain Decomposition
- suits well for parallel computing
- a good parallelization strategy
- Object-Oriented Programming Technique
- flexible and efficient sequential simulators
- can be used in subdomain solves -- main ingredient of DD

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- A simulator-parallel model
- Each processor hosts an arbitrary number of subdomains
- balance between numerical efficiency and load balancing
- One subdomain is assigned a sequential simulator
- Flexibility -- different types of grids, linear system solvers, preconditioners, convergence monitors etc. are allowed for different subproblems
- Domain decomposition on the level of subdomain simulators!

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- Reuse of existing sequential simulators
- Data distribution is implied
- No need for global data
- Needs additional functionalities for exchanging nodal values inside the overlapping region
- Needs some global administration

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- An add-on library (SPMD model)
- Use of object-oriented programming technique
- Flexibility and portability
- Simplified parallelization process for end-user

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- Parameter Interface
solution method or preconditioner, max iterations, stopping criterion etc

- DD algorithm Interface
access to predifined numerical algorithme.g.CG

- Operation Interface (standard codes & UDC)
access to subdomain simulators, matrix-vector product, inner product etc

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- Subdomain Simulator -- a generic representation
- C++ class hierarchy
- Interface of generic member functions

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SubdomainSimulator

SubdomainFEMSolver

NewWSimulator

- Class SubdomainSimulator - generic representation of a sequential simulator.
- Class SubdomainFEMSolver - generic representation of a sequential simulator using FEM.
- A new sequential wave simulator that fits in the framework is
- readily extended from the
- existing sequential simulator,
- also being a subclass of
- SubdomainFEMSolver.

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WaveSimulator

- Algorithmic efficiency
- efficiency of original sequential simulator(s)
- efficiency of domain decomposition method
- Parallel efficiency
- communication overhead (low)
- coarse grid correction overhead (normally low)
- synchronization overhead
- load balancing
- subproblem size
- work on subdomain solves

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- Fixed number of subdomains M=16.
- Subdomain grids from partition of a global 41x41x41 grid.
- Simulation over 32 time steps.
- DD as preconditioner of CG for the Laplace eq.
- Multigrid V-cycle as subdomain solver.

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- Number of subdomains equal to number of processors

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*ForP=2 parallel BiCGStab is used.

- Efficient solution of elliptic boundary value problems
- Parallelization based on DD
- Introduction of a simulator-parallel model
- A generic framework for implementation
http:www.nobjects.com

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