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Copyright©2004 VINAS Co.,Ltd.

2004/03/25

Super Matrix SolverFast & Robust Sparse Matrix Solver

Product Introduction

April 2004

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Table of Contents

What is Super Matrix Solver (SMS)?………………...…3What is Super Matrix Solver -AMG?……………...…...4Benefits of using Super Matrix Solver …………….......5Issues in matrix calculation………………………….…6Performance Super Matrix Solver-AMG………..…......7Performance Super Matrix Solver-AMG…….………...8Special Feature of Super Matrix Solver (SMS)……….9Specifications of Super Matrix Solver -AMG(1)…….10Specifications of Super Matrix Solver -AMG(2)…….11Specifications of Super Matrix Solver -AMG(3)…….12Using Super Matrix Solver-AMG..…………..………13SMS-AMG Product Development Schedule…………14SMS-AMG V2………………………………………..15SMS-AMG Product Series………………………........162004-2006 SMS Product Development Schedule….....17

Super Matrix Solver -AMG Major Customers in JapanNational Laboratories, Universities…………………...18Industry.......………………...………………………....19

Super Matrix Solver License Fee Schedule…………..…20Performance and Application Examples….……………..21Benefits of Super Matrix Solver…………………….….22Application of SMS-AMG

Concrete strength study ………………..……………...23Nuclear power plant fuel rods fluid force vibration analysis…...243D plastic mold flow analysis……...………………….252D plastic mold flow analysis…………………..….….26Electromagnetic and fluid coupled analysis…....……...27Incompressible fluid flow analysis……………..…...…28Electromagnetic analysis…...………………………….29

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Jointly developed with National Aerospace Laboratory of JapanAcceleration technology for iterative matrix solution methods. It can be incorporated into various existing solution methods to accelerate and stabilize process of matrix calculation.

Conventional iterative methods

Super Matrix Solver

Super Matrix Solver has an optimization algorithm to accelerate and stabilize solution process of iterative methods such as SOR and AMG.

Algorithm of SMS

What is Super Matrix Solver (SMS)?

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What is Super Matrix Solver-AMG?

AMG method stands for Algebraic Multi Grid method, which is an iterative method developed during 1980-1990 based on the latest calculation theories. It is a fast calculation method, but not widely used because of following problems.

Difficult to program. Needs advanced mathematical knowledge. Difficult to find out best combination of parameters due to large number of parameters to set.

Matrix solver based on AMG method and speeded-up and stabilized by Super Matrix Solver technology of VINAS. One module of Super Matrix Solver library.

General AMG method

Super Matrix Solver-AMG

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Benefits of using Super Matrix Solver

Speed-up the calculation process by numerical analysis programs without enhancing computer hardware resourcesObtain converged solution for problems that have never been solved by conventional methods due to divergence and robustness of high-speed calculation methods

Obtain accurate solutions for very large models in a stable way

Minimize manual setting of parameters

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Enhance Calculation Speed Contributes directly to speeding-up

the whole CFD solver process・ Increase convergence rate・ Implement parallel processingImprove robustness and convergence rate・Prevent divergence of calculation・Prevent stagnation of convergence

Issues in matrix calculationUsability enhancements・Reduce number of parameters to set・Introduce easier way of convergence judgement

Improve accuracy of Numerical AnalysisEssential in large calculations・Eliminate cancellation of significant digits・Reduce accumulation of errors

Faster, larger and cheaper computers

Enhancements in CAD/CAE/CFD software

Wider field of application of CAE/CFD software

Target of CAE/CFD

Larger models

More complex models

Demand for higher accuracy

Transient simulations

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Performance of Super Matrix Solver-AMG

Logarithm of relative residual vs. CPU time for calculation

Advantages of Super Matrix Solver-AMG

Based on AMG method and is made faster and more robustMuch faster than CG methods Problems that have never been solved by other methods can be robustly solvedVery stable convergence during calculation

Super Matrix Solver-AMG and other solver methods

CPU-TIME

log(|r|/|b|)

Calculation of 1.1 million unknowns

Super Matrix Solver-AMG

GPBi-CG

SOR

Approx.. 1200 sec.

50 sec

* GPBi-CG is relatively fast and robust among CG methods

Normal AMG

300 sec

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Performance of Super Matrix Solver-AMG

Results of convergence history of simultaneous linear equation by Super Matrix Solver-AMG, normal AMG, normal SOR, and GPBi-CG solvers are compared above. Reduction of common logarithms of relative residuals (ordinate) versus CPU time (abscissa) is plotted. Calculations were stopped when relative residuals reached 1.0 x 10e-4.* GPBi-CG is a solution method known by its speed and stability among CG methods.

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Special Feature of Super Matrix Solver（SMS）

1) Fast calculation

2) High stability and high convergence rate

3) Reduced parameter setting

4) Not liable to error accumulation ■ Keeps accurate solution ■ Suitable to very large calculations

5) Comparatively less iterations needed for large calculation

0

5

10

15

20

25

0 10 20 30 40

(x 10,000)

Iteration

Number of unknowns(proportional to number of elements)

Proportional increase

Super Matrix Solver

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Specifications of Super Matrix Solver-AMG (1)

Type of matrix Sparse matrixSupports calculation of structured and unstructured grid in CFD and other analyses

Supports calculation of asymmetric matrix

Zero diagonal elements Can not calculate

Number of unknowns No limitation

Type of variables (single) and double precision versions

Specifications of Super Matrix Solver-AMG

Specifications may change without prior notice.

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Input parameters (*1) Left side coefficient matrix (A), Right side constant vector (b) , convergence criteria, maximum iterations, ...

Output data Solution vector(X), Relative residual, Number of performed iterations, …

Error messages Warnings and error messages are returned as return values (calculation information, system information, …).




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Supported systems Windows, Unix(SGI/IRIX, ...）Linux and parallel versions planned* Consult us for details

Provided as Library (source is not open)

Documents Manual (explains data format, parameters, application development, and so on)

Sample data Sample program using Super Matrix Solver-AMG

Field of application Can be used for solving linear equation system in CFD, structural, electromagnetic and other analyses.

License management Managed by physical address of computerRuns on designated machine only




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Super Matrix Solver Can be incorporated with Fortran, C and C++ programs. Shown below is an example of using Super Matrix Solver-AMG from a Fortran program.

rtc=smsamgd(x,abrs,nstp,ad,alu,b,lnt,lnd,nd,ns,mstep,eps, lu0sw)

Using Super Matrix Solver-AMG

Specifications may change without prior notice. For the details about the parameters, refer to the product manual.

NOTE：rtc = return code lnt = column indices (j) of non-diagonal, non-zero elements in matrix A (i, j)x = solution of unknown x lnd = numbers of non-diagonal, non-zero elements in each row of matrix Aabrs = achieved accuracy nd = dimension of matrix Anstp = actual number of iteration ns = number of non-diagonal, non-zero elements in matrix Aad = values of diagonal elements of matrix A mstep = maximum iterationalu= values of non-diagonal, non-zero elements eps = required accuracy/convergence level in relative residual

in matrix A lu0sw = control switch for initial values of unknown vector xb = values of right-hand side constant vector b

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SMS-AMG Product Development Schedule

Version 3

Version 1

Version 4

Version 5

CFD;Electro-magnetic Analysissingle degree

of freedom

Structure Analysis

solid elementCFD

multi degree of freedom

Structure Analysis

shell element

Electro-magneticAnalysis

ｅｄｇｅ element

Application

Version 2 Parallel Version

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SMS-AMG V2

=

2

1

2221

1211

bb

PU

aaaa

1P

2P5P

3P

4P4U1U

2U5U

3U

[ ][ ] [ ]bPa =

1P

2P5P

3P

4P

SMS－AMG１ SMS－AMG２

SMS-AMG V2 will support much wider field of applications

Existing SMS focuses on single degree of freedom (DOF) problems of u, v, w, p and so on. V2 will address multiple DOF problems.

Multiple DOF problems such as structural analysis problems can be robustly solved.

Residual in log scale

0 50 100 150-6

-5

-4

-3

-2

-1

0αSMS-AMG2

ICCG

- Solid element structural analysis

- 3 variables (three spatial components of displacement)

- Approx. 60,000 unknowns

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SMS-AMG Product Series

•X•X•Parallel

•Ver4

•CFD coupled•Structure analysis

•CFD segregated

•Application

•OK•X•Solid elements in structure analysis

•X•X•Shell elements in structure analysis

•X•X•Complex value

•X•X•Edge elements in electromagnetic analysis

•X•X•Dense matrix

•X•X•Zero elements in diagonal

•Released•Released•Status

•OK•X•Multi variables

•OK•OK•Sparse matrix

•Ver X•Ver3•Ver2•Ver1

02/10/2004ISSUE03

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O ctober ‘03 - M arch ‘041 .) β -version of SM SV 3 for shell com pleted (end of M arch 2004)2 .) Porting of P -IC C G for each p latform com pleted (for JM A G )3 .) D evelopm ent o f P -IC C G M PI Version (resu lt o f co llaborative research w ith JA E R I: Japan A tom ic E nergy R esearch Institu te)4 .) Product com pletion o f D irect M ethod So lver

A pril ’04 – Sep tem ber ‘041 .) Product com pletion o f SM SV 3 for Shell E lem ent (end of June 2004)2 .) Product com pletion o f P -IC C G M PI Version3 .) D evelopm ent and p roduct com pletion of SM SV 4 for E dge E lem ent

O ctober ’04 – M arch ‘051 .) D evelopm ent and p roduct com pletion of SM S-A M G Parallel Version (SM P Version)2 .) Product C om pletion o f B E M Solver

A pril ’05 – Sep tem ber ’05D evelopm ent and p roduct com pletion of SM S-A M G Parallel Version (M PIVersion)

O ctober ’05 – M arch ’061 .) P roduct com pletion o f E igen Value So lver

2004-2006 SMS Product Development Schedule (as of 11/25/03)

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■National Laboratories, Universities

Super Matrix Solver Major Customers in Japan (February 2004 Issue 5)

Central Research Institute of Electric Power Industry CFD

Fukui University CFD

Hokkaido University Structural Strength (Breakage)

Japan Aerospace Exploration Agency CFD

Japan Atomic Energy Institute CFD

Japan Nuclear Cycle Development Institute CFD

Kyusyu Institute of Technology Electromagnetic Analysis

Kyusyu University CFD

National Institute of Materials Science Structural Analysis (Molecular Level)

Okayama University CFD

Osaka University Structural Analysis

Port and Airport Research Institute CFD

Saitama Institute of Technology CFD

Shizuoka University CFD

The Institute of Physical and Chemical Research CFD

The University of Tokyo CFD

Tokyo Institute of Technology CFD

Yokohama National University, etc CFD

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■Industry

Super Matrix Solver Major Customers in Japan (February 2004 Issue 5)

CANON INC. CFDCRC Solutions Corp. CFDFuji Research Institute Corp. CFDHitachi, Ltd. CFDKajima Corporation Structural AnalysisMatsushita Electric Industrial Co., Ltd. Electromagnetic AnalysisMazda Motor Corporation Aerodynamics AnalysisNikon Corporation Electromagnetic AnalysisPlamedia Corporation Plastic Mold Flow AnalysisRicoh Co., Ltd. Electromagnetic AnalysisShimizu Corporation Groundwater Flow AnalysisSumitomo Metal Industries, Ltd. CFDToray Industries, Inc. Plastic Mold Flow AnalysisToshiba Tec Corporation CFD (Two-phase flow)μ-Tech Inc., etc Electromagnetic Analysis

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Super Matrix Solver License Fee Schedule(Effective as of September 1, 2003)DRAFT

1. Product Name SMS-AMG-V1, V2

2. Serial license fee (perpetual license) US$(consumption tax not included) CPU#Product# 1 2-4 5-16 17-32 33-48 49-64 65-128 129-256 257-512 513-SMSAMG32-STD

(32bit CPU) 4,000 8,000 16,000 24,000 32,000 42,000 62,000 80,000 96,000 Contact usSMSAMG32-STD

(642bit CPU) 6,000 12,000 24,000 36,000 48,000 63,000 93,000 120,000 160,000 Contact us

3. Media and Shipment charge US$(consumption tax not included)

Product Number Item FeeSMS-CD-ROM (Additional order) CD-ROM 100

[NOTE]

1. Numbers of CPU’s in above tables are numbers of CPU’s that are physically connected either in SMP type or in cluster type environment.

2. Maximum size of calculation models - Up to five million degrees of freedom is supported. In order to run models with larger size, use SMS-Professional series.

3. The software is provided as an executable. 4. The license is a node lock type, which means that the software works on a fixed machine.5. License fees shown above are fees for perpetual licenses with a free of charge 3 months support service

(includes bug fix updates and operation support). Annual support service after the 3 months is available at a fee. 6. Licenses must not be resold or assigned to any third parties other than its original licensee.

Resale is only permitted under OEM or Distributor contract. 7. Licenses are valid for the latest version available at the time of purchase only.

When a new version becomes available and a user wants to use it, the user must purchase license(s) for the new version. 8. Contact us for information about academic licenses. 9. Consumption tax is not included in the tables above.

10. Information provided above, including fees, may change without prior notice.

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Performance and

Application Examples

Performance and

Application Examples

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100 200 300

5

0 0 20 40 60 80 100

-4

-3

-2

-1

0

1

Benefits of Super Matrix SolverFaster numerical analysis calculationsFaster numerical analysis calculations

Highly accurate solutions can be obtained for very large scale problems

Highly accurate solutions can be obtained for very large scale problems

Solutions to previously insoluble problems

Solutions to previously insoluble problems

Minimized manual settings of calculation parameters

Minimized manual settings of calculation parameters

Large problems

Higher hardware specs

More difficult problems

条件数の悪化

rtc=smsamgd(solution, coefficient matrix, right-hand side vector, number of iterations, normalization, convergence criterion, reduction)

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Application of SMS-AMG

Saitama Institute of Technology

[Field of application]: CFDNuclear power plant fuel rods fluid force vibration analysis

Comment from Hisashi Hishida, Ph. D,Saitama Institute of Technology

[Calculation issues]Trial of some large models took a few months to be solved by conventional iterative methods, but their calculation was significantly speeded-up by SMS-AMG.

Calculations to achieve same accuracy （10-2）were speeded up 50 to 100 times.

114,000 unknowns

SOR

SMS-AMG

(sec)

Nuclear Power Plant Fuel RodsFluid Force Vibration Analysis

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Plamedia Corporations

Table Comparison of CPU time

Model

Direct solver

Iterative solver

SMS AMG-

Eqn. 52357Non0s. 4525296

Eqn. 143386Non0s. 12144782

283.4 sec

81.9 sec

Error (Memory over)

100.8 sec

(Monitor：Pentium IV PC, 2.53GHz, 2Gb)

6 MPa

Fig. Pressure distribution 0

3D plastic mold flow analysis

3.5 times faster calculations


Previously insoluble problems solved in high speed

Study of heat and flow inside a spiral mandrel die and quantitative analysis of residence time profile by using 3D finite element analysis

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（1/6）

>1/40

2D plastic mold flow analysis

Plamedia Corporation6 times faster than famous Direct method



[Field of application]Mold flow analysis of handheld phone

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0.00

50.00

100.00

150.00

200.00

250.00

300.00

u v w p

98,000 cells

PCG

AMG

SMS-AMG

DIRECT

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

u v w p

98,000 cells

10-6

10-10

Application of SMS-AMGCRC Solutions Corporation

5 20

10

2

Inlet

Outlet

Inflow: Constant x-velocity(18.2)Outflow: Gauge pressure = 0Kinematic viscosity: 1.5x10-5

Density: 1.0Boundaries: Cylindrical surface has non-slip, other surfaces have slip condition.

SMS-AMG is integrated into FINAS/CFD code

Calculation time for u,v,w,p

Calc. Time for different convergence level

（Unable to calculate w/ AMG）

Electromagnetic and fluid coupled analysis

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No. of elements: Approx. 40,000Reynolds number: 1,000Unknowns: 40,000Conv. Criterion: 1E-4Method: MAC method

10 times faster than before

Incompressible fluid flow analysis

0

1

2

3

4

5

6

7

8

9

10

SOR SMS-AMG

SOR

SMS-AMG

SOR SMS-AMGC

alcula

tion t

ime (

hr)

Approx. 10 hours

Better convergence performance was observed at the beginning of calculation where instability is often observed. One of SMS-AMG’s advantages is that it can carry out calculation more stably than SOR even with a larger time step.

Comment from user

The Institute of Space and Astronautical Science (ISAS) of Japan



Approx. 1 hour

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Electromagnetic analysis

PC version of SMS-AMG from VINAS is capable of delivering its high performance in electromagnetic analyses through our performance tests. Speed of calculation is far above what can be expected from conventional solution methods. With appropriate computer hardware, SMS-AMG is expected to be able to handle calculations with DOF exceeding 2 million.

User Statement

MU TECH Co., Ltd.Four times faster in calculation of 1 Million DOF problem


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IEEJ’s 3D static electromagnetic model for verification purpose

Iron core and coil model, one-eighth symmetric part

Hardware specifications HP workstation x1100 / Pentium4 2GHz/ RAM 786MB, Hard disk 32GB