School Fundamentals and practice of finite elements

Roscoff, april, 2017

part 3 : Advanced XLiFE++

E. LunévilleUnité de Mathématiques Appliquées

POEMS, ENSTA ParisTech

eXtended Library

of Finite Elements

in C++

2

Dealing with BEM Coupling FEM-BEM, FEM-IR Dealing with transient problem Dealing with non linear problem XLiFE++ in future

Outline

Advanced XLiFE++

3Advanced XLiFE++

Helmholtz scattering problem

Integral representation formulasHelmholtz kernels

Limits when x goes to (G is singular)

BEM Helmholtz 3D

4

Jump relations ( any Helmholtz solution in )

limit on

Simplest integral equation for the exterior Dirichlet problem

take

(EFIE)

well posed if k is not an eigenvalue of the Dirichlet interior problem

Advanced XLiFE++

BEM Helmholtz 3D

5XLiFE++ introduction

Mesh MS(Sphere(_center=Point(0.,0.,0.), _radius=1., _hsteps=0.1), _triangle, 1,_gmsh);Domain Gamma = MS.domain(0);

Deal with integral equations in XLiFE++ (EFIE, sphere)

• Mesh the unit sphere with triangles

• Define space (Lagrange P0) and unknown

Space V(Gamma, P0,"V");Unknown phi(V0,"phi"); TestFunction psi(phi,"psi");

• Get the Helmholtz kernel

Real k=5;Parameters pars(k,"k"); Function F(f,pars);Kernel G=Hemholtz3dKernel(k);

f is chosen as an incident plane wave

Complex f(const Point& P, Parameters& pars = defaultParameters){

Real k = pars("k");return -exp(i_*k*P(1));

}

GMSH mesh (3018 triangles)

BEM Helmholtz 3D

6

IntegrationMethods collects integration methods to apply • to different parts of a Kernel (_allFunction, _regularPart, _singularPart)• to different configurations of elements

IntegrationMethods ims(_SauterSchwabIM, 4, 0.,_defaultRule, 5, 2.,_defaultRule, 3);

Methods for double integral with singular Kernel (or singular part of Kernel)• _SauterSchwabIM (quadrature) for Kernel with 1/r singularity (Laplace, Helmholtz, Maxwell 3D)• _DuffyIM (quadrature) for Kernel with log r singularity (Laplace, Helmholtz 2D)• _LenoirSalles3dIM (exact for P0-P1) for Laplace3D kernel or singular part of Helmholtz 3D• _LenoirSalles2dIM (exact for P0-P1) for Laplace2D kernel or singular part of Helmholtz 2D

Sauter-Schwab (order 4 on segment) method when elements are adjacent (81 quadrature points)best quadrature rule of order 5 for element at relative distance <=2 (49 quadrature points)else best quadrature rule of order 3 (9 quadrature points)

Advanced XLiFE++

BEM Helmholtz 3D

7

• Define forms and compute matrix (in dense storage) and rhs

Bilinearform a = intg(Gamma, Gamma, phi*G*psi);LinearForm b = intg(Gamma, F*psi);TermMatrix A(a,"A"); TermVector B(b,"B");

Advanced XLiFE++

TermVector X=gmresSolve(A,B);

• Represent solution on the sphere and save it to file "U.vtu"

IntegrationMethods imsh(LenoirSalles3dIR(),_singularPart, theRealMax,QuadratureIM(Gauss_Legendre,4),_regularPart, theRealMax);

TermVector U = integralRepresentation(phi, Gamma, intg(Gamma,G*phi,imsh), X, "SL0");saveToFile("U", U,_vtu);

Use exact integration method (Lenoir-Salles) for singular part and standard quadrature for regular part

• Solve using GMRES

Paraviewcell vizualization (P0)

Paraviewpoint vizualization(filter Cell Data To Point Data)

real part of the scattered field

BEM Helmholtz 3D

8Advanced XLiFE++

Doing BEM in P1, change only the space

Space V(Gamma, P1,"V");real part of the scattered field

P1 computation

Representing the total field on planes(create new mesh, domain and space)

Real c=0;Rectangle Rx(_v1=Point(c,-3.,-3.), _v2=Point(c,3.,-3.),

_v4=Point(c,-3.,3.), _hsteps=0.1);Mesh Px(Rx,_triangle,1,_gmsh);Domain dompx=Px.domain(0);Space Vx(dompx,P1,"Vx"); Unknown ux(Vx,"ux");TermVector Upx = integralRepresentation(ux, dompx,

intg(Gamma,G*phi, imsh), Ut, "Upx");saveToFile("Upx",Upx-TermVector(ux,dompx,F),_vtu);...

Compute the total field on Gamma (gives 0!)

TermVector Ut = U-TermVector(phi,Gamma,F);real part of

the total field

modulus of the total field

BEM Helmholtz 3D

9Advanced XLiFE++

BEM Helmholtz 3DOther standard BEM formulations

(MFIE)

(CFIE)

Bilinearform a = 0.5*intg(Gamma, phi*psi)+ intg(Gamma, Gamma, phi*ndotgradx(G)*psi);

LinearForm b = -intg(Gamma, (_n|ginc)*psi); //ginc : grad of uinc

Bilinearform a = 0.5*intg(Gamma, phi*psi)+ intg(Gamma, Gamma, phi*ndotgradx(G)*psi)+ eta * intg(Gamma, Gamma, phi*G*psi);

LinearForm b = -intg(Gamma, (_n|ginc)*psi) //ginc : grad of uinc+ eta * intg(Gamma, (uinc*psi) ;

10Advanced XLiFE++

BEM Maxwell 3DIndirect EFIE

well posed if k is not an eigenvalue of the Dirichlet interior problem

(Stratton-Chu)

Change the space – use Hdiv conform space

Space RT(Gamma, RT_1, "RT");Unknown U(RT,"U"); TestFunction V(U,"V");

Change forms

BilinearForm a = (1/k) * intg(Gamma,Gamma, div(U)*G*div(V), ims)- k * intg(Gamma,Gamma, (U*G)|V, ims);

TermMatrix A (a,"A");TermVector B(intg(Gamma, fi|V), "B");TermVector X = directSolve(A,B);

ComplexVector Ei(const Point& P, Parameters& pars)

{ RealVector pol(3,0.); pol(1)=1.;Point dir(0.,0.,1.) ;Real k = pars("k");return pol*exp(i_*k * dot(P,dir));}

Data function Ei

11Advanced XLiFE++

BEM Maxwell 3DChange integral representation on planes using Stratton-Chu

IntegrationMethods ims2(_defaultRule,10, 1.,_defaultRule, 5);Unknown ux3(Vx,"ux", 3); // new vector unknown !TermVector E = -(1/k)*integralRepresentation(ux3, dompx, intg(Gamma,grad_x(G)*div(U),ims2), X)

-k *integralRepresentation(ux3, dompx, intg(Gamma,G*U,ims2), X);

use standard quadrature (order 10 if close to Gamma else order 5)

real part of Ex

12Advanced XLiFE++

FEM-IRA drawback of BEM is the computation of singular integralsTo avoid this computation, a method consists in coupling FEM and Integral representation

Helmholtz with Neumann condition

Equivalent problem

13Advanced XLiFE++

FEM-IR

XLiFE++ can deal with such formulation, diffraction by a sphere

Sphere S1(_center=Point(0.,0.,0.), _radius=1., _hsteps=0.1,_side_names="Gamma");

Sphere S2(_center=Point(0.,0.,0.), _radius=1.2, _hsteps=0.12, _domain_name="Omega", _side_names="Sigma");

Mesh mC(S2-S1,_tetrahedron,1,_gmsh);Domain Omega = mC.domain("Omega");Domain Gamma =mC.domain("Gamma"), Sigma =mC.Domain("Sigma");Gamma.setNormalOrientation(outwardsDomain ,Omega);Sigma.setNormalOrientation(outwardsDomain ,Omega);

Sigma

GammaOmega

Space V(Omega, P2, "V");Unknown u(V,"u"); TestFunction v(u,"v");

use standard P2 Lagrange space

BilinearForm a = intg(Omega,grad(u)|grad(v))-(k*k)*intg(Omega,u*v)+eta*intg(Sigma, u*v)+ intg(Sigma, Gamma, u*ndotgrad_x(ndotgrad_y(G))*v)+ eta*intg(Sigma, Gamma, u* ndotgrad_y(G)*v);

TermMatrix A(a,"A");

Bilinear form

be careful with normal orientation and signs in variational form

14Advanced XLiFE++

FEM-IRcompute right hand side using bilinear form and solve

BilinearForm af = intg(Gamma, u*v) + intg(Sigma, Gamma, u*ndotgrad_x(G)*v)+ eta*intg(Sigma, Gamma, u*G*v);

TermMatrix Af(af);Function dnf(dnui, pars); dnf.associateVector(_n);TermVector Dnf(u, Gamma, dnf);TermVector B = Af*Dnf;TermVector X=directSolve(A,B);

Function ui(f,pars); IntegrationMethods imr(_defaultRule,10,1.,_defaultRule, 5);Rectangle Ry(_v1=Point(-3.,0.,-3.), _v2=Point(3.,0.,-3.),

_v4=Point(-3.,0.,3.), _hsteps=0.1);Rectangle Rz(_v1=Point(-3.,-3.,0.), _v2=Point(3.,-3.,0.),

_v4=Point(-3.,3.,0.), _hsteps=0.1);Mesh Py(Ry,_triangle,1,_gmsh), Pyz(Rz,_triangle,1,_gmsh);Pyz.merge(Py); //merge meshesDomain dompyz=Pyz.domain("main domain");Space Vyz(dompyz,P1,"Vyz"); Unknown uyz(Vyz,"uyz");TermVector Upyz = integralRepresentation(uyz, dompyz, intg(Gamma,G*D,imr))

- integralRepresentation(uyz, dompyz, intg(Gamma,ndotgrad_y(G)*X,imr));TermVector Uinc(uyz,dompyz,ui), Ut=Upyz+Uinc;

Do integral representation on plane y=0 and z=0

Complex dnui(const Point& P, Parameters& pa = defaultParameters)

{ Real k = pa("k"), x=P(1);RealVector n = pa.getVector(_n);return -i_*k*exp(i_*k*x)*n(1);}

modulus of the total field

15Advanced XLiFE++

FEM-BEM

Solving a diffraction problem with penetrable obstacle requires FEM computationin the obstacle coupled with a method for solving the exterior problem : many are possible : FEM-DtN, FEM-IR, FEM-BEM, ...

BEM (EFIE)

FEM

Multiple unknowns problemXLiFE++ is designed for this purpose !!!

16Advanced XLiFE++

FEM-BEM

2D example

Disk D1(_center=Point(0.,0.), _radius=1., _hsteps=0.1,_domain_name="Omega",_side_names="Gamma");

Square C(_origin=Point(-4,-4),_length=8,_hsteps=0.1, _domain_name="S");

Mesh mC(C+D1,_triangle,1,_gmsh);Domain Omega = mC.domain("Omega");Domain Gamma =mC.domain("Gamma");Domain S =mC.domain("S");

Space V(Omega, P1, "V"); Unknown u(V,"u"); TestFunction v(u,"v");Space L(Gamma, P0, "L"); Unknown l(L,"l"); TestFunction t(l,"t");

use standard P1 Lagrange space for FEM and P0 Lagrange for BEM

Real k=5; Parameters pars(k,"k");Kernel G=Helmholtz2dKernel(k);Function fk2(k2,pars); Function finc(uinc,pars);IntegrationMethods ims(Duffy,10,0., defaultQuadrature,6,2.,defaultQuadrature,4);BilinearForm a = intg(Omega, grad(u)|grad(v))-intg(Omega, k2*u*v)-*intg(Gamma, l*v)

+ 0.5*intg(Gamma, u*t) - intg(Gamma, Gamma, u*ndotgrad_y(G)*t)+ intg(Gamma, Gamma, l* G*t);

LinearForm b=intg (Gamma, finc *t ) ;

Bilinear form

S

Gamma

17Advanced XLiFE++

FEM-BEM

Solve system ant do integral representation on S

TermMatrix A(a,"A"); TermVector B(b,"B");TermVector X=directSolve(A,B);TermVector Ui=X(u); IntegrationMethods imr(LenoirSalles2dIR(),_singularPart, theRealMax,

QuadratureIM(defaultQuadrature,4),_regularPart, theRealMax);Space VS(S,P1,"VS"); Unknown uS(VS,"uS");TermVector US = integralRepresentation(uS,S,intg(Gamma, ndotgrad_y(G)*Ui, imr))

- integralRepresentation(uS,S,intg(Gamma, G*X(l), imr));TermVector UtS = US + TermVector(uS,S,finc)

real part of the total field modulus of the total field

Example

18Advanced XLiFE++

FEM-BEM : a funny application

Diffracted portrait done for the French Science Fair ("Fête de la Science"2017)• take a picture with a camera• insulate the face and adjust contrast, ...• pixelize and map pixels to sound velocities• compute diffraction by plane waves using FEM-BEM in XLiFE++• visualize the result

19Advanced XLiFE++

What about time dependent problemsXLife ++ deals with all the time dependent problems and none!

no particular stuff related to time dependent problems, standard stuff allows to deal with most.

Solve wave equation in bounded domain using standard leap-frog scheme

Variational form of the leap-frog time discretisation:

Matrix form after spatial discretisation in a FE space Vh

20Advanced XLiFE++

What about time dependent problems

Square sq(_origin=Point(0.,0.), _length=1, _nnodes=70);Mesh mesh2d(sq, triangle, 1, structured);Domain omega=mesh2d.domain("Omega");Space V(omega, P1, "V", true); Unknown u(V, "u"); TestFunction v(u, "v");TermMatrix A(intg(omega, grad(u)|grad(v)),"A"), M(intg(omega, u*v), "M");TermVector G(intg(omega, g*v), "G");TermMatrix L; ldltFactorize(M,L);Real c=1, dt=0.004, dt2=dt*dt, cdt2=c*c*dt2, t=dt;Number nbt=200;TermVectors U(nbt); TermVector zeros(u, omega, 0.); U(1)=zeros; U(2)=zeros;for (Number n=2; n<nbt; n++, t+=dt)

U(n+1)=2.*U(n)-U(n-1)-factSolve(L, cdt2*(A*U(n)) - dt2*h(t)*G);saveToFile("U", U, vtu);

mesh & domain

space & unknowncomputematrices & vectors

initialize time loopallocate a TermVectorsto store states a each tn

time loop, computingstate at tn+1

save all the states

t = 0.16s t = 0.32s t = 0.48s t = 0.64s

21Advanced XLiFE++

What about non linear problemsSame answer as for time dependent problems : XLife ++ deals with all the non linear problems and none!

no particular stuff related to non linear problems, standard stuff allows to deal with some.

Finding some invisible obstacles in a waveguide (work of A.S Bonnet-BenDhia and A. Bera)

In the unperturbed waveguide ( ρ= 0), there are modal solutions

finite number of propagative modes

other are evanescent modes

only two mode, that is a plane wave

22Advanced XLiFE++

What about non linear problems

Assume low frequency regim and incident plane wave

perfect invisibilitynon reflectivity

expression of coefficients

perturbation theory

23Advanced XLiFE++

What about non linear problems

Algorithm

implemented in XLiFE++ with standard stuffmore complex cases have been dealt (T=1 case, two propagative modes)

24XLiFE++ introduction

Invisibility in a waveguide

invisibility of a collection of penetrable obstacles

Simple obstacle inducing no reflection

25XLiFE++ introduction

XLiFE++ in future

To come soon edge and face element on quadrangle and hexahedron (work in progress) Buffa-Christiansen element (work in progress) HMatrix for Maxwell BEM (not working but easy upgrade) tools for DDM (MPI interface)

To come next Fast Multipole Method (interface to FastMMLib) management of gmsh new features in mesh generation wrapper to PETSC-SLEPC (non linear eigenvalue problems)

To come later discontinuous Galerkin HMatrix factorization augmented geometry (CAO) high frequency tools (PTDG)