Improving the Solution Method in Wanda

Leonard Huijzer

Delft University of Technology, The Netherlands

March 6, 2018

Water Hammer

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Water Hammer

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Water Hammer

2

Structure

1 Model

2 Current Situation

(a) Robustness(b) Performance(c) Maintainability

3 Research Approach

(a) Robustness(b) Performance(c) Maintainability

4 Summary and Approach

5 Planning

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Model – Steady State Flow

Liquid module quantities:

• Energy head: H = pρg + z

• Volumetric flow rate: Q = Av

Example: Small System (incompressible, viscous)

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Model – Steady State Flow

Example: Small System (incompressible, viscous)

H1 = c1QA + Q1 − Q2 = 0

QA = 0HA = H1

HA = H2

H2 − H3 =λL

8A/O

Q2|Q2|A2g

Q2 = Q3

HB = H3

HB = H4

QB = 0QB + Q3 + Q4 = 0

H4 = c2

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Model – Steady State Flow

Apply Newton-Raphson method:

f(u(k+1)) ≈ f(u(k)) + J(u(k))�u(k+1) − u(k)

�= 0

where u(k) = [Q(k)1 H

(k)1 . . . Q

(k)n H

(k)n ]� and J(u(k)) is the Jacobian

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Model – Steady State FlowApply Newton-Raphson method:

Example: Small System

0 1 0 0 0 0 0 0 0 0 0 01 0 1 0 −1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 1 0 −1 0 0 0 0 0 0 0 00 0 0 −1 0 1 0 0 0 0 0 00 0 0 0 −c3 1 0 −1 0 0 0 00 0 0 0 1 0 −1 0 0 0 0 00 0 0 0 0 0 0 1 0 −1 0 00 0 0 0 0 0 0 0 0 −1 0 10 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 1 0 1 00 0 0 0 0 0 0 0 0 0 0 1

Q1

H1

QA

HA

Q2

H2

Q3

H3

QB

HB

Q4

H4

=

c10000c400000c2

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Model – Transient FlowViscous , compressible in pipes

∂v

∂t+ g

∂H

∂x+

λ

8A/Ov |v | = 0

∂H

∂t+

c2

g

∂v

∂x= 0

c = pressure wave speedSolution method:

1 Solve internal pipe points atΔt/2

2 Solve internal pipe points atΔt

3 Solve rest of system + pipeboundaries at Δt usingNewton-Raphson

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Model – Transient Flow

Example: Sewage System

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Current Situation – Robustness

Example: Steady Flow H

Now:

H2 − H3 =λL

8A/O

Q2|Q2|A2g

has an infinite number of solutions. Fix: Prescribe H on A or B .

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Current Situation – Robustness

Example: Transient Flow H

Component phases can cause trouble.Fix: Take H from previous time-step.

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Current Situation – Robustness

Example: Steady State Q

H1 = c1H1 = HA

QA + Q1 + Q2 = 0QA = 0H2 = HA

H2 = c2

Fix??Can also happen in transient flow simulations.Furthermore, if c1 �= c2 then contradiction!

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Current Situation – Performance

What are we working with?

(a) Drinking water (b) Noord-Holland 1

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Current Situation – Performance

What are we working with?

(a) Drinking water (b) Noord-Holland 1

Small (but many), asymmetric, sparse and banded� LU-decomposition

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Current Situation – PerformanceCurrent solution method

1 Determine component ordering2 For each time step

(a) If phase change: detect if and where H is undetermined; apply fix(b) Solve internal pipe points(c) Solve system using Newton-Raphson � IMSL matrix solver

Matrix: solve using LU-decomposition with Markowitz pivoting:Select pivot which minimises fill-in

(a) 2D Laplacian (b) L (c) U

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Current Situation – Maintainability

Some systems are unsolvable:

• Underdetermined

• Contradictory

Lead to singular matrices

Why is this a problem?:

• International Mathematics and Statistics Library (IMSL) matrixsolver crashes or loops

• Limited troubleshooting

• Paid license

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Research Goals

• Robustness: prevent or handle singular matrices

• Performance: no concessions

• Maintainability: open source library with permissive license

Goal:Find and implement numerical library which satisfies these requirements

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Research Approach – RobustnessGoal: prevent singular matrices• Physical model• Structurally singular systems

Definition

Let M ∈ Rn×n. M is called structurally singular if every N ∈ Rn×n,with Nij = 0 whenever Mij = 0, is singular.

H1 = c1H1 = HA

QA + Q1 + Q2 = 0QA = 0H2 = HA

H2 = c2

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Research Approach – Robustness

Let the matrix M ∈ R6×6 bedefined by

Mij =

�1, if variable j in equation i0, otherwise

H1 = c1H1 = HA

QA + Q1 + Q2 = 0QA = 0H2 = HA

H2 = c2

M =

1 0 0 0 0 01 0 1 0 0 00 1 0 1 0 10 0 0 1 0 00 0 1 0 1 00 0 0 0 1 0

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Research Approach – Robustness

6

5

4

3

2

1

Q2

H2

QA

HA

Q1

H1

H1 = c1H1 = HA

QA + Q1 + Q2 = 0QA = 0H2 = HA

H2 = c2

M =

1 0 0 0 0 01 0 1 0 0 00 1 0 1 0 10 0 0 1 0 00 0 1 0 1 00 0 0 0 1 0

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Research Approach – Robustness

Goal: detect ’singular’ matricesDeterminant not useful in finite precision.

Definition: Condition Number

Let M ∈ Rn×n. The condition number of M is defined as

κ(M) = �M� · �M−1�

Use estimation techniques.

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Research Approach – Robustness

Goal: detect ’singular’ matricesDeterminant not useful in finite precision.

Singular Value Decomposition

Let M ∈ Rn×n. The SVD is given by

M = UΣV�

where Σ = diag(σ1,σ2, . . . ,σn) and σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0.

Note that: κ2(M) =σmax

σminAlternatives: QR-decomposition, . . .

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Research Approach – Performance

Test cases

(a) Noord-Holland 1 (b) Noord-Holland 2

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Research Approach – Performance

Test cases

H-detect IMSL solver Matrix build Miscellaneous0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Perc

enta

ge o

f tim

e

Total time: 68s

(a) Noord-Holland 1

H-detect IMSL solver Matrix build Miscellaneous0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Perc

enta

ge o

f tim

e

Total time: 1654s

(b) Noord-Holland 2

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Research Approach – PerformancePerformance improvements:• Fill-in reduction by component ordering; now Breadth-First Search

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Research Approach – PerformancePerformance improvements:• Fill-in reduction by component ordering; now Breadth-First Search• Fill-in reduction via matrix reordering: e.g., Cuthill-McKee

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Research Approach – Performance

Performance improvements:

• Fill-in reduction by component ordering; now Breadth-First Search

• Fill-in reduction via matrix reordering: e.g., Cuthill-McKee

• Fill-in reduction via pivoting: e.g., Markowitz

At each step minimise

(r(k)i − 1)(c

(k)j − 1)

where r(k)i = nnz(M(k)(i , :)) and c

(k)j = nnz(M(k)(:, j))

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Research Approach – Performance

Performance improvements:

• Fill-in reduction by component ordering; now Breadth-First Search

• Fill-in reduction via matrix reordering: e.g., Cuthill-McKee

• Fill-in reduction via pivoting: e.g., Markowitz

• (LU-decomposition + condition number estimation vs.SVD,QR ,. . . )

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Research Approach – Performance

Other potential improvements:

• Newton-Raphson alternatives: Quasi-Newton, Picard iteration

• Algorithm that detects undetermined nodes

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Research Approach – Maintainability

Requirements:

• Open source

• Permissive license

• (Free)

Candidates:

• Linear Algebra Package (LAPACK)

• PLASMA, MAGMA

• Multi-frontal Massively Parallel Solver (MUMPS)

Which offers best performance?

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Summary and ApproachSummary:• The Wanda model: steady and transient flow• Current situation: robustness, performance, maintainability

Approach:1 Prevent singular matrices:

• detect structural singularities• physical model

2 Matrix solver performance vs. other routines3 Implement LAPACK

• condition number estimation• rank-revealing decomposition• Matrix reordering• Evaluate performance

4 If necessary, implement PLASMA, MAGMA, MUMPS5 Other improvements

• Algorithm which detects undetermined nodes• Newton-Raphson alternatives• . . .

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Planning

1 March: Detect structural singularities, matrix solver performancevs. other routines (using test cases)

2 April: Replace IMSL by LAPACK, detect singular matrices,improve matrix ordering

3 May: Condition number estimation, rank-revealingdecompositions, evaluate performance and robustness

4 June: Test PLASMA, MAGMA, MUMPS

5 July: Report draft, other improvements

6 August: Finish report, presentation

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Improving the Solution Method in Wanda

Leonard Huijzer

Delft University of Technology, The Netherlands

March 6, 2018

Bibliography

Anderson, E., Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz,A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen (1999). LAPACK Users’Guide. Third. Philadelphia, PA: Society for Industrial and Applied Mathematics.

Golub, G. and C. Van Loan (1996). Matrix Computations. Johns Hopkins Studies in theMathematical Sciences. Johns Hopkins University Press. isbn: 9780801854149.

Hydrodynamica van Leidingsystemem (2015). Deltares.

IMSL Fortran Math Library (2014). RogueWave Software.

Multifrontal Massively Parallel Solver. User’s Guide (2016).

WANDA 4.5 User Manual (2017). Deltares.

Wylie, E. and V. Streeter (1978). Fluid transients. Advanced book program v. 1977.McGraw-Hill International Book Co.

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