symplegma , a jvm implementation for numerical methods in...

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SymJava

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Symplegma, a JVM implementation for

numerical methods in computational mechanics.

Christos G. Panagiotopoulos

Institute of Applied & Computational MathematicsFoundation of Research and Technology–Hellas

FOSSCOMM 2018, 13-14 OctoberFree and Open Source Software Communities Meeting

Computer Science DepartmentUniversity of Crete

The Symplegma project

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What is Symplegma?


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Symplegma: a computational mechanics framework

and beyond

• Java code developed for personal research activity.

• Main orientation, education and research.

The spatiotemporal path of the project:

• 2008-2010, Aristotle University of Thessaloniki,Thessaloniki, Greece

• 2010-present, University of Seville, Seville, Spain

• 2010-2011, Charles University of Prague, Prague, Czechia

• 2013, GEOMEC Engineering, Stavanger, Norway

• 2014-present, Foundation of Research and Technology,Heraklion, Crete

• 2017-2018, Technical University of Crete, Chania, Crete

• 2017-present, Technological Educational Institute ofCrete, Rethymno, Crete


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Symplegma: a computational mechanics framework

and beyond


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Symplegma on the net


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Computational mechanics and numerical computing

From International Association for Computational Mechanics website

Computational Mechanics (CM) is the development and applicationof numerical methods and digital computers to the solution ofproblems posed by Engineering and Applied Science with theobjectives of understanding and harnessing the resources of nature.

Interdisciplinary

CM involves different fields of expertise. Its three pillars aremathematics, computer science, and mechanics. Computational fluiddynamics, computational thermodynamics, computationalelectromagnetics, computational solid mechanics are some of themany specializations within CM.


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Numerical methods

Mathematics

Ordinary and Partial differential equations, linear algebra andnumerical analysis.

Methods

FEM, FDM, and BEM are the most popular numerical methodsused in order of dominance. In solid mechanics FEM are farmore prevalent, whereas in fluid mechanics, thermodynamics,and electromagnetism, FDM are almost equally applicable. TheBEM is in general less popular, but has a niche in certain areas(e.g., acoustics, fracture).


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Java?

General

Java is a general purpose, computer programming language, gained

widespread popularity. Appealing features of Java include: clean and

simple syntax, high level of maintainability of Java code, portability,

automatic memory management, multithreading and networking

capabilities, wealth of libraries and tools.

Science

Early interest in Java from the scientific communities to determine the

level of performance offered by Java, and its suitability for developing

scientific software. The Java Grande Forum (JGF) was an initiative to

encourage interest in Java amongst the scientific and HPC communities.


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Why and for what reason?

Java

Java is considered as an alternative, and not subsitute, language fornumerical, scientific and engineering computations. FORTRAN,C/C++ are the dominants with Python and Julia being suppementary.Java should be considered both for the language and the Java VirtualMachine, while lot of significant languages running on the JVM havebeen arised (e.g. Groovy, Scala, Clojure, Jython, JRuby, etc.).

Symplegma & SDE

Symplegma and the respective platform SDE are considered tobecome an integrated computational mechanics platform with mainorientation research and education. Substitution of commercial oropen source code capable to handle huge problems of industry is notconsidered.


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The Symplegma components

Components used:

• Java, Groovy

• Climax, SDE

• Apache Commons Math, JAMA, JBLAS

• JLaTeXMath

• Futureye, SymJava

• . . . among other . . .


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The Climax packages

Some packages developed as part of Climax :

• jbem: Boundary element method components

• jfem: Finite element method components

• gendomain: A general purposes domain

• geom: Utilities for geometric entities

• sympmesher: A simple mesher

• jdem (draft): Discrete element method

• structuraldynamics: Structural dynamics and vibrations

Symplegma may turn out to be a pocketknife-like tool. . .


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SDE: An integrateddevelopment environment forSymplegma components


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Description of SDE environment

The SDE environment consists of three segments

(a) the text area (input)

(b) the communication area (output)

(c) the area of graphical visualization (output)

key: simplicity


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Plots - PlotFrame


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Plots - PlotFrame - jlatexmath


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The java source code, classesand libraries


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The code skeleton

Pre-processing

• Declare variables

• Define domains

• Define nodes

• Define element

• Define other necessaryentities

Processing

• Declare algorithms

• Set analysis’ parameters

• Do the analysis

Post-processing

• Manipulating results

• Visualize

• Plotting


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jfem: The finite elementmethod implementation


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Beam elastodynamics

Euler-Bernoulli

EI∂4v

∂x4+ A

∂2v

∂t2= q(x, t)

Timoshenko

κGA

(

∂ψ

∂x−∂2v

∂x2

)

+ ρA∂2v

∂t2= q(x, t)

κGA

(

∂v

∂x− ψ

)

+ EI∂2ψ

∂x2− I

∂2ψ

∂t2= 0.


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Climax (Groovy) Code

1 import j f em .∗ ;23 // d e f i n e a fem domain4 dom=theUn i v e r s e . FEMDomain( ) ;56 L=0.1; NNodes=200;78 // d e f i n e ma t e r i a l9 mat= new E l a s t i cM a t e r i a l ( 1 , 193000000000 , 0 . 25 , 7850 . 0)

1011 // d e f i n e c ros s−s e c t i o n s12 s e c t= new C r o s s S e c t i o n ( 1 , 0 . 0046 , 0 . 00096)1314 // d e f i n e nodes15 ( 1 . . NNodes ) . eachdom . putNode (new Node ( i t , i t ∗L/NNodes ) )1617 // d e f i n e e l ement s18 (1. .<NNodes ) . eachdom . putE lement (new EBeam2d ( i t , dom . getNode ( i t ) ,dom . getNode

( i t +1) ,mat , s e c t ) )1920 // d e f i n e a n a l y s i s21 t h eA n a l y s i s = new E i g e nAna l y s i s (dom , 3 )22 t h eA n a l y s i s . a n a l y s e ( )2324 // p l o t e igen−modes25 theGP . p lotDe form (−4 ,1.0)


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Beam elastodynamics, first bending eigenmode


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Beam elastodynamics, second bending eigenmode


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Beam elastodynamics, third bending eigenmode


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Plane stress elasticity: plate with hole

• Modulus of elasticityE=200GPa

• Poisson ratio ν=0.25

• Lx=Ly=10.0cm

• Radius r=0.5m

• Dirichlet boundaryd = 0.1cm

• Navier-Lame equations ofequilibrium


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Plate with hole: mesh


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Plate with hole: deformed


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Plate with hole: σxx stress


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Plate with hole: σyy stress


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Plate with hole: τxy stress


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Plate with hole: Von Misses stress


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FEM implementation features (non exhaustive list)

Element

• Single dof

• Truss

• Beam

• Plane stress/strain

• Constraint elementMaterial

• Elastic

• Inelastic (plasticity,hardening)

Algorithm

• Linear

• Non linear(Newton-Raphson)

Analysis

• Quasi-static

• Transient dynamic (timedomain)

• Eigenanalysis


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jbem: The boundary elementmethod implementation


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BEM potential problem

Laplace Equation and Harmonic Functions.

x

y

ΓDΓD

∇2u(x, y) = 0 in Ω

u = f on ΓD

∂u

∂n=∂f

∂non ΓN

f = e4x2−(2y−1)2 sin (4x(2y − 1))


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BEM potential problem (discretization)


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BEM potential problem (u-distribution)


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BEM potential problem (∂u∂n-distribution)


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BEM elasticity problem (discretization)


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BEM elasticity problem (deformed)


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BEM elasticity problem (deformed internal-1)


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BEM elasticity problem (deformed internal-2)


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Contact mechanics problem with BEM

E (u) =

N∑

i=1

Ei(ui) =

N∑

i=1

∫

Ωi

1

2C(i)ei(ui):ei(ui) dΩ.

W (t, u) =

N∑

i=1

∫

ΓN

ui(x, t) · piN(x, t) dS

Π(t, u) =

E (u)− W (t, u) if u = uD on ΓD,

[[u]]n ≥ −gn on ΓC,

∞ elsewhere,


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Contact problem with BEM


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BEM implementation features (non exhaustive list)

Geometry

• One dimensions

• Two dimensions

• Three dimensions

Element

• Point 1 node

• Line 2-3 nodes

• Quad 4-9 nodes

Material

• Elastic (iso-, orthotropic)

• Viscous

• Potential

Analysis

• Quasi-static

• Transient elastodynamics

• Viscoelastic

• Multiple domains

• Contact


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Climax general domain andusage of SDE as a numerical

computing platform


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The generalized domain

• The gendomain is an entity that mainly consists of nodeswith degrees of freedom and respective values on severaltime steps.

• There is a direct connection of gendomain with thegraphics area.

• We present it here through finite differences method(FDM) examples solving the wave equation.

∂2u

∂t2= c2

∂2u

∂x2x ∈ Ω, t ∈ (0, T ]

u(x, 0) = u0(x),∂

∂tu(x, 0) = v0(x) x ∈ Ω

u(x, t) = u x ∈ ΓD, t ∈ (0, T ]

∂u(x, t)

∂x= q x ∈ ΓN, t ∈ (0, T ]


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Time evolution of deformation (00/10)


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Wave propagation in 2D acoustic medium

For the circular membrane problem the wave equation in (r, φ)polar coordinates

1

c2∂2u

∂t2=∂2u

∂r2+

1

r

∂u

∂r+

1

r2∂2u

∂φ2

under approrpiate boundary and initial conditions. Aftersubstitution of finite difference equations we may arrive at:

un+1i,j = c2∆t2

[ 1

2ri∆r(uni+1,j − uni−1,j) +

1

∆r2(uni−1,j + uni+1,j

− 2uni,j) +1

r2i∆φ2(uni,j−1 + uni,j+1 − 2uni,j)

]

− un−1i,j + 2uni,j

The Courant number in that case is C = 2c∆t∆r∆φ

stabilitycondition

C ≤ 1 → ∆t ≤∆r∆φ

2c.


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Wave propagation on circular membrane (01/08)


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Coupling

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The structuraldynamicspackage as a toolbox foreducational purposes


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Classes as entities of structural dynamics

Classes of some structural dynamics entities:

• sdof the single degree of freedom system

• sdofSeries a system of masses connected throughsprings in series

• shearframe a shear frame multistorey frame

• pendulum a multiple pendulum (currently up to twomasses mey be considered)

All these classes implement the interface contraption whichmainly has the function SelfPortrait and Motion both forconnection and print on the graphics area.


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Double pendulum

Kinetic energy

T =1

2(m1 +m2)l

21θ

21 +m2l1l2θ1θ2 cos (θ1 − θ2) +

1

2m2l

22θ

22.

Potential energy,

V = −(m1 +m2)gl1 cos θ1 −m2gl2 cos θ2 + const.

Equations of motion through Lagrange’s equations:

d

dt

(

∂T

∂qi

)

−∂T

∂qi+∂V

∂qi= 0

Solving using a pendulum’s function that utilizesClassicalRungeKuttaIntegrator (Runge-Kutta) fromCommons Math of Apache.


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Pendulum’s motion (01/09)


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Other packages than Climax. . . and more. . .


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FuturEye Java PDE solverwith FEM


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Introduction

FuturEye

FuturEyea is a Java based Finite Element Method (FEM)Toolkit, providing concise, natural and easy understandingprogramming interfaces for users who wish to develop researchand/or engineering FEM algorithms for Forward and/or InverseProblems.

ahttps://github.com/yuemingl/Futureye_JIT

Weak form abstraction

One of the most appealing features is that of the weak formclasses for approximating PDEs’ strong forms. Different weakforms may be involved in different techniques for solving thesame PDE problem with FEM.


https://github.com/yuemingl/Futureye_JIT

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Weak forms of PDEs

https://en.wikipedia.org/wiki/Weak_formulation


https://en.wikipedia.org/wiki/Weak_formulation

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Solve Poisson’s equation

Find u ∈ H1(Ω) such that

−∆u(x, y) = f(x, y), (x, y) ∈ Ω

u(x, y) = 0, (x, y) ∈ ∂Ω

f(x, y) = −2(x2 + y2) + 36


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PDE potential problem in SDE


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Coupling

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PDE FuturEye implementation features (nonexhaustive list)

Function

• Constant

• Single variable

• Combination

• Composition

• Derivative

• Several variables

• Vector function

Weak form

• Laplace

• Convection-diffusion

• Navier-Stokes

• . . .

Operators

• div

• grad

• curl


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SymJava A Java library forfast symbolic-numeric

computation


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Introduction

SymJava

SymJavaa is a Java library for symbolic-numeric computation.Java bytecode is generated at runtime for symbolic expressionswhich make the numerical evaluation really fast.

ahttps://github.com/yuemingl/SymJava

Utilization

While it is not included in SDE by default, its jar file might beloaded from menu bar (Load→Add jar).


https://github.com/yuemingl/SymJava

The

Symplegma

project

Symplegma

SDE

Climax

jfem

jbem

gendomain

strdyn

More

FuturEye PDE

SymJava

Coupling

Conclusions

A differentiation example

1 // h t t p s :// g i t hub . com/ yueming l /SymJava2 import s t a t i c symjava . symbo l i c . Symbol . ∗ ;3 import symjava . bytecode . BytecodeFunc ;4 import symjava . symbo l i c . ∗ ;5

6 Expr exp r = x + y∗z ;7 p r i n t l n ( exp r ) ; //x + y∗ z8

9 Expr expr2 = exp r . subs ( x , y∗y ) ;10 p r i n t l n ( expr2 ) ; //yˆ2 + y∗ z11 p r i n t l n ( expr2 . d i f f ( y ) ) ; //2∗ y + z12

13 Func f = new Func ( ” f1 ” , ( expr2 . d i f f ( y ) ) ) ;14 p r i n t l n ( f ) ; //2∗ y + z15

16 BytecodeFunc func = f . toBytecodeFunc ( ) ;17 p r i n t l n ( func . app l y (1 ,2 ) ) ; // 4 .0


The

Symplegma

project

Symplegma

SDE

Climax

jfem

jbem

gendomain

strdyn

More

FuturEye PDE

SymJava

Coupling

Conclusions

Coupling of differentnumerical methods and/ordifferent physics domains


The

Symplegma

project

Symplegma

SDE

Climax

jfem

jbem

gendomain

strdyn

More

FuturEye PDE

SymJava

Coupling

Conclusions

Domain Decomposition Methods (e.g., DtN)

Initialize ubfi on interface Γbf = Γfb and iterate while p > tol:

1 Solve BEM → tbfi

2 tbfi → f

fbi

3 Solve FEM → ufbi

4 ubfi+1 = (1− θ)ubfi + θu

fbi

5 Compute some norm p of difference between ubfi+1 and ubfiwhere θ a relaxation parameter usually in [0, 1].


The

Symplegma

project

Symplegma

SDE

Climax

jfem

jbem

gendomain

strdyn

More

FuturEye PDE

SymJava

Coupling

Conclusions

Educational activity: past

Experience gained:

• Previously developed software, dynasoft and edusoft, atAristotle University of Thessaloniki, for courses such“Strength of Materials” and “Structural Dynamics”.

• Course on “Computational Mechanics” at the Departmentof Production Engineering & Management, TechnicalUniversity of Crete.

• Course on “Computational Acoustics” at the Departmentof Music Technology and Acoustics Engineering of theTechnological Educational Institute of Crete.

• Course on “Dynamics, vibration and control of structures”at the Department of Production Engineering &Management, Technical University of Crete.


The

Symplegma

project

Symplegma

SDE

Climax

jfem

jbem

gendomain

strdyn

More

FuturEye PDE

SymJava

Coupling

Conclusions

Educational activity: current

Experience to be gained (?):

• Reformation of the course on “Computational Acoustics”at the Department of Music Technology and AcousticsEngineering of the Technological Educational Institute ofCrete.

• Course on “Applied Mathematics” at the Department ofMusic Technology and Acoustics Engineering of theTechnological Educational Institute of Crete.

• Course on “Structural Acoustics” at the Department ofMusic Technology and Acoustics Engineering of theTechnological Educational Institute of Crete.


The

Symplegma

project

Symplegma

SDE

Climax

jfem

jbem

gendomain

strdyn

More

FuturEye PDE

SymJava

Coupling

Conclusions

Conclusions

• Improvement and optimization of existed code

• Advanced coupling strategies between domains of differentnumerical methods (e.g., FEM-BEM) and/or physics (e.g.,Fluid-Structure)

• Advanced plotting features

• Further development of the 3D graphics area (JavaFX)


symplegma , a jvm implementation for numerical methods in...

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