M´ ethode d’´ el´ ements de fronti` ere Marc Bonnet UMA (Dept. of Appl. Math.), POems, UMR 7231 CNRS-INRIA-ENSTA 32, boulevard Victor, 75739 PARIS cedex 15, France [email protected]´ Ecole doctorale MODES Methodes num´ eriques avanc´ ees, 28 mars 2013 http://uma.ensta.fr/ mbonnet/enseignement.html Marc Bonnet (POems, ENSTA) M´ ethode d’´ el´ ements de fronti` ere 1 / 183
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Methode d’elements de frontiere
Marc Bonnet
UMA (Dept. of Appl. Math.), POems, UMR 7231 CNRS-INRIA-ENSTA32, boulevard Victor, 75739 PARIS cedex 15, France
4. The fast multipole method (FMM) for the Laplace equationMultipole expansion of 1/rThe single-level fast multipole methodThe multi-level fast multipole method
5. The fast multipole method (FMM) for elastostatics
6. The fast multipole method for elastodynamics
7. Other acceleration methodsExponential representation of 1/rFMM using equivalent sourcesClustering and low-rank approximationsKernel-independent acceleration via kernel interpolationAdaptive cross approximation
8. Preconditioning
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 2 / 183
Review of boundary integral equation formulations
1. Review of boundary integral equation formulationsElectrostaticsLaplaceElastostaticsFrequency-domain wave equations
2. Review of classical BEM concepts3. The GMRES iterative solver4. The fast multipole method (FMM) for the Laplace equation
Multipole expansion of 1/rThe single-level fast multipole methodThe multi-level fast multipole method
5. The fast multipole method (FMM) for elastostatics6. The fast multipole method for elastodynamics7. Other acceleration methods
Exponential representation of 1/rFMM using equivalent sourcesClustering and low-rank approximationsKernel-independent acceleration via kernel interpolationAdaptive cross approximation
8. Preconditioning
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 3 / 183
Review of boundary integral equation formulations Electrostatics
1. Review of boundary integral equation formulationsElectrostaticsLaplaceElastostaticsFrequency-domain wave equations
2. Review of classical BEM concepts
3. The GMRES iterative solver
4. The fast multipole method (FMM) for the Laplace equation
5. The fast multipole method (FMM) for elastostatics
Review of boundary integral equation formulations Electrostatics
Electrostatics
I Well-known, and simple, physical setting
I Allows to introduce important concepts of integral equation formulationswith a clear physical meaning
I Said concepts will generalize to other settings (elasticity, electromagnetics...)
I Also helpful later for a physical understanding of the fast multipole method(FMM)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 5 / 183
Review of boundary integral equation formulations Electrostatics
Electrostatics: discrete charged particles
I Coulomb interaction force:
F12 =q1q2
4πε
1
r 2r12
q1r
q2r12
I Electrostatic field:
F12 = q2E12, E12 =q1
4πε
1
r 2r12
I Electrostatic potential:
E12 = −∇2V , V =q1
4πε
1
r
(with ε: permittivity of the medium (material constant))
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 6 / 183
Review of boundary integral equation formulations Electrostatics
Electrostatics: continuous charge distributions
I Continuous charge distribution: dq = % dV :
E(x) =1
4πε
∫V
%(ξ)
r 2r dVξ r = ξ−x, r = ‖r‖, r = r/r
I Gauss theorem:
div E =%
ε
I Poisson equation (Gauss theorem with E = −∇V ):
∆V +%
ε= 0
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 7 / 183
Review of boundary integral equation formulations Electrostatics
Electrostatics: continuous charge distributions
Proof of Gauss theorem:∫V
div E dV =
∫∂V
E·n dS
=1
4πε
∫V
∫∂V
1
r 2(r·n(x)) dSx
%(ξ) dVξ
=1
4πε
∫V
Θ(ξ)%(ξ) dVξ
where Θ(ξ) is the solid angle of the (closed) surface ∂V from origin ξ:
Θ(ξ) = 4π (ξ ∈V ), Θ(ξ) = 0 (ξ ∈R3 \ V )
Hence: ∫V
div E dV =
∫V
%
εdV
Since this is true for any domain V , one has div E = %/ε, i.e. the Gauss theoremholds.
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 8 / 183
Review of boundary integral equation formulations Electrostatics
Electrostatic volume potential
The electrostatic volume potential results from the superposition of electric fieldsgenerated by elementary charges % dV distributed in a volume V :
V[%,V ](x) =1
4πε
∫V
%(ξ)
rdVξ
As already seen, V[%,V ] satisfies the Poisson equation:
∆V[%,V ] +%
ε= 0 (x∈V ), ∆V[%,V ] = 0 (x∈R3 \ V )
Properties of electrostatic volume potentials
I The volume integral is weakly singular (i.e. singular, but integrable) forx∈V , so that V[%,V ] is well-defined inside the charged domain V : one has
dVx = r 2 dr dΘ = O(r 2)
I V[%,V ] is continuous everywhere, and in particular across the boundary ∂V .
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 9 / 183
Review of boundary integral equation formulations Electrostatics
Electrostatic single-layer potential
The electrostatic single-layer potential results from the superposition of electricfields generated by elementary charges % dS distributed on a surface S :
S[%, S ](x) =1
4πε
∫S
%(ξ)
rdSξ
S[%, S ] is harmonic outside of S :
∆S[%, S ] = 0 (x∈R3 \S)
Properties of electrostatic single-layer potentials
I The surface integral is weakly singular (i.e. singular, but integrable) for x∈ S ,so that S[%,S ] is well-defined on the charged surface V : one has
dSx = r dr dθ = O(r)
I S[%, S ] is continuous everywhere, and in particular across S .
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 10 / 183
Review of boundary integral equation formulations Electrostatics
Electrostatic double-layer potentialThe electrostatic double-layer potential is the limiting case of the superposition oftwo single-layer potentials of (i) arbitrary close supports S±, (ii) opposite chargedensity, (iii) finite dipolar moment q:
D[q,S ](x) = limh→0S[%h,S
+](x) + S[−%h,S−](x)
with
limh→0[h%h](x) = q(x)
(note: q is analogous to a concentrated moment)
n
x+x
x−x+ = x + h
2n
x− = x− h2n
S+
S−
Performing the limit h→ 0, one finds:
D[q,S ](x) = − 1
4πε
∫S
1
r 2r·n(ξ) q(ξ) dSξ
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 11 / 183
Review of boundary integral equation formulations Electrostatics
Electrostatic double-layer potential
D[q,S ](x) = − 1
4πε
∫S
1
r 2r·n(ξ) q(ξ) dSξ
Properties of electrostatic double-layer potentials
I D[q,S ] is harmonic outside of S :
∆D[q,S ] = 0 (x∈R3 \S)
I The surface integral is weakly singular (i.e. singular, but integrable) for x∈ S ,so that D[q,S ] is well-defined on the charged surface V .This is not obvious and stems from
dSx = r dr dθ = O(r) and1
r 2r·n = O(
1
r)
I D[q,S ] is discontinuous across S , with
D[q,S ](x+)−D[q,S ](x−) = q(x)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 12 / 183
Review of boundary integral equation formulations Electrostatics
Electrostatic potentials: comments
(a) Electrostatic potentials have a clear physical meaning as the potential fieldsassociated with volume, surface or dipolar charge distributions.
(b) Electrostatic potentials, as mathematical constructs, define harmonic fields forarbitrary choices of supports V ,S and densities %, q.
(c) As we will shortly see, any harmonic function can be expressed in terms ofsuch potentials.
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 13 / 183
Review of boundary integral equation formulations Laplace
1. Review of boundary integral equation formulationsElectrostaticsLaplaceElastostaticsFrequency-domain wave equations
2. Review of classical BEM concepts
3. The GMRES iterative solver
4. The fast multipole method (FMM) for the Laplace equation
5. The fast multipole method (FMM) for elastostatics
Review of boundary integral equation formulations Elastostatics
Reciprocity identity and integral representation
I Additive decomposition of strain into elastic and initial (e.g. thermal, plastic,visco-plastic) parts (assuming infinitesimal strain):
ε = εE + εI where σ = C :εE
I Constitutive equation (C: fourth-order tensor of elastic moduli):
σ = C : (ε− εI)
I For isotropic elasticity (µ: shear modulus, ν: Poisson ratio):
Cijk` = µ[ 2ν
1− 2νδijδk` + (δikδj` + δi`δjk)
]I Governing field equation for unknown displacement field u(ξ):
Cijab(ua,bj − εI
ab,j) + bi = 0 (ξ ∈Ω)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 27 / 183
Review of boundary integral equation formulations Elastostatics
Elastostatic fundamental solution
Kelvin fundamental solution: unit point force applied at x∈R3 alongk-direction in unbounded elastic medium, i.e.:
CijabUka,bj + δ(· − x)δik = 0 (ξ ∈R3)
Uki (x, ξ) =
1
16πµ(1− ν)
1
r
[ri rk + (3− 4ν)δik
]Σk
ij(x, ξ) = − 1
8π(1− ν)
1
r 2
[3ri rk rj + (1− 2ν)(δik rj + δjk ri − δij rk)
]x, ξ ∈R3
T ki (x, ξ) = Σk
ij(x, ξ)nj(ξ)
( r = |ξ−x| , ri = (ξi −xi )/r )
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 28 / 183
Review of boundary integral equation formulations Elastostatics
Reciprocity identity and integral representationGoverning field equation for unknown displacement field u(ξ):∫
Ω
Cijab(ua,bj − εI
ab,j) + bi
× Uk
i (x, ξ) dVξ = 0 (1)
Governing field equation for fundamental solution:∫Ω
CijabUk
a,bj + δ(ξ − x)δik× ui (ξ) dVξ = 0 (2)
Performing (1)-(2) and invoking the divergence identity, one obtains the integralrepresentation formula of the displacement:
uk(x) =
∫∂Ω
Uki (x, ξ)ti (ξ)− T k
i (x, ξ)ui (ξ)
dSξ
+
∫Ω
Uki (x, ξ)bi (ξ) + Σk
ij(x, ξ)εI
ij(ξ)
dVξ
I Partially unknown contribution of ∂Ω (BC + unknown trace)I Known contribution of Ω (if εI is known beforehand)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 29 / 183
Review of boundary integral equation formulations Elastostatics
Displacement boundary integral equation
Limiting process as x∈Ω −→ z∈ ∂Ω in integral representation:
uk(x) =
∫∂Ω
Uki (x, ξ)ti (ξ)− T k
i (x, ξ)ui (ξ)
dSx
+
∫Ω
Uki (x, ξ)bi (ξ) + Σk
ij(x, ξ)εI
ij(ξ)
dVξ
Γε
xnsε
εz
Ω
Non-integrable singularity of Tk(x, ξ):
I Limit to the boundary approach
I Direct approach using exclusion neighbourhood of z
I Indirect regularization approach
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 30 / 183
Review of boundary integral equation formulations Elastostatics
Displacement boundary integral equationIntegral equation, singular form:
1
2uk(x) + P.V.
∫∂Ω
T ki (x, ξ)ui (ξ) dSx −
∫∂Ω
Uki (x, ξ)ti (ξ) dSx
=
∫Ω
Uki (x, ξ)bi (ξ) + Σk
ij(x, ξ)εI
ij(ξ)
dVξ
Integral equation, regularized form:∫∂Ω
T ki (x, ξ)[ui (ξ)− ui (x)]− Uk
i (x, ξ)ti (ξ)
dSx
=
∫Ω
Uki (x, ξ)bi (ξ) + Σk
ij(x, ξ)εI
ij(ξ)
dVξ
I Both forms require u∈C 0,α (otherwise process x∈Ω→ z∈ ∂Ω breaks down)I Numerical implementation based on (well-documented) singular element
integration methods.I Boundary-only formulations in the absence of body forces and initial strains.I Treatments (double / multiple reciprocity methods) sometimes allow to
convert domain integrals with b, εI into boundary integrals.Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 31 / 183
Review of boundary integral equation formulations Elastostatics
Volume, single-layer and double-layer elastic potentials
I Volume potential (prescribed body forces):
Vbk [b,Ω](x) =
∫Ω
Uki (x, ξ)bi (ξ) dVξ
(displacement field created in R3 by b given on Ω)I Volume potential (prescribed initial strains):
Vεk [εI,Ω](x) =
∫Ω
Σkij(x, ξ)εI
ij(ξ) dVξ
(displacement field created in R3 by εI given on Ω)I Single-layer elastic potential:
S[ϕ, ∂Ω]k(x) =
∫∂Ω
Uki (x, ξ)ϕi (ξ) dSx
I Double-layer elastic potential:
D[ψ, ∂Ω]k(x) =
∫∂Ω
T ki (x, ξ)ψi (ξ) dSx
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 32 / 183
Review of boundary integral equation formulations Elastostatics
I Half-space with free surface (Mindlin, 1936), Boussinesq as special case:exact solutions (within half-space idealization):−→ Soil mechanics and geotechnics;−→ Contact mechanics (Hertz solution, Galin formulae)
I Two perfectly-bonded half spaces (Rongved, 1955) – closed form
I Elastic layer between two parallel planar free surfaces (Benitez and Rosakis1987) – integral transform
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 34 / 183
Review of boundary integral equation formulations Frequency-domain wave equations
1. Review of boundary integral equation formulationsElectrostaticsLaplaceElastostaticsFrequency-domain wave equations
2. Review of classical BEM concepts
3. The GMRES iterative solver
4. The fast multipole method (FMM) for the Laplace equation
5. The fast multipole method (FMM) for elastostatics
valid if u satisfies a radiation condition at infinity:I Integral form:
limR→∞
∫SR
(G,n(x, ξ)u(ξ)− G (x, ξ)u,n(ξ)
)dSξ = 0
I Local form, sufficient, known as Sommerfeld condition:
∇u ·x− iku = o(‖x‖−1) ‖x‖ −→ ∞
(Sommerfeld is known to imply u = o(1), i.e. decay of u at infinity)
The radiation condition is satisfied by G (x, ·), and consequently also by
• The fundamental solution
• Volume, single-layer, double-layer potentials
• Integral representation formula
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 40 / 183
Review of boundary integral equation formulations Frequency-domain wave equations
Scattering of incident waves by hard obstacles
I Governing equations (hard obstacle[s])
∆u + k2u = 0 in Ω
∇u ·n = 0 on ∂Ω (no normal velocity, i.e. hard obstacle)
I Known incident wave (or ’free-field’) uF; radiation conditions not assumed(e.g. plane wave):
∆uF + k2uF = 0 (in R3)
I Decomposition:
u = uF + v = incident + scattered, + radiation conditions for v
Ω∂
Ω
uF
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 41 / 183
Review of boundary integral equation formulations Frequency-domain wave equations
Scattering of incident waves by hard obstacles
I Scattered field verifies integral equation (by virtue of radiation cnds):
c(x)v(x) +
∫∂Ω
G,n(x, ξ)v(ξ)− G (x, ξ)v,n(ξ)
dSξ = 0 (a)
I Free-field verifies integral equation for interior of scatterer:
[c(x)− 1]uF(x) +
∫∂Ω
G,n(x, ξ)uF(ξ)− G (x, ξ)uF
,n(ξ)
dSξ = 0 (b)
I Simplified integral equation formulation (a)+(b) in terms of total field:
c(x)u(x) +
∫∂Ω
G,n(x, ξ)u(ξ) dSξ = uF(x)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 42 / 183
Review of boundary integral equation formulations Frequency-domain wave equations
Fictitious eigenfrequencies
BIE formulations for exterior problems break down when ω is an eigenfrequencyfor a certain interior problem.
Example:
I Direct BIE formulation for exterior Neumann problem (scattering by rigidobstacle):
c(x)u(x) +
∫∂Ω
G,n(x, ξ)u(ξ) dSξ = uF(x) (a)
I Indirect BIE formulation (using a double-layer potential representation) forinterior homogeneous Dirichlet problem (using same normal as (a)):
c(x)ψ(x) +
∫∂Ω
G,n(x, ξ)ψ(ξ) dSξ = 0 (b)
I (b) has non-trivial solutions if ω is a Dirichlet eigenvalue.
I Therefore, so does (a) as the governing integral operator is the same
Remedies include:(i) enforcing an extra set of integral identities at interior points;(ii) combining (with complex coefficients) two BIE formulations having differenteigenvalues (see treatment in Pyl, Clouteau, Degrande 2004)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 43 / 183
Review of boundary integral equation formulations Frequency-domain wave equations
Scattering of incident waves by penetrable obstacles
I Penetrable inclusion (ρ?(ξ), c?(ξ) and with definitions β = ρ/ρ?, γ = c/c?):
(∆ + k2)u = 0 (in Ω \ B) (∆ + γ2k2)u? = 0 (in B)
u = u?, u,n = βu?,n (on ∂B)
I Domain integral equation of Lippman-Schwinger type (proof: combinereciprocity identities written on B for u? and on R3 \B for u−uF):
u(x) +
∫B
[(β − 1)∇G (x, ξ)∇u(ξ) + (1− βγ2)k2G (x, ξ)u(ξ)
]dVξ = uF(x)
c ( ), ( )* *ρ xx uF
Ω
B
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 44 / 183
Review of boundary integral equation formulations Frequency-domain wave equations
Linear elastodynamics
Governing field equation:
div (C :ε[u]) + ρω2u + b = 0 (in Ω)
Fundamental solution (full space):
Uki (x, ξ) =
1
k2Sµ
[(δqsδik − δqkδis)
∂
∂xq
∂
∂ξsG (‖x−ξ‖; kS) +
∂
∂xi
∂
∂ξkG (‖x−ξ‖; kP)
]G (z ; k) =
eikz
4πz, k2
S =ρω2
µ, k2
P =1−2ν
2(1−ν)k2
S
Radiation conditions for unbounded media, local form:
σ[uP]·x− iρωcP = o(‖x‖−1)
σ[uS]·x− iρωcS = o(‖x‖−1)
‖x‖ −→ ∞
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 45 / 183
Review of classical BEM concepts
1. Review of boundary integral equation formulationsElectrostaticsLaplaceElastostaticsFrequency-domain wave equations
2. Review of classical BEM concepts3. The GMRES iterative solver4. The fast multipole method (FMM) for the Laplace equation
Multipole expansion of 1/rThe single-level fast multipole methodThe multi-level fast multipole method
5. The fast multipole method (FMM) for elastostatics6. The fast multipole method for elastodynamics7. Other acceleration methods
Exponential representation of 1/rFMM using equivalent sourcesClustering and low-rank approximationsKernel-independent acceleration via kernel interpolationAdaptive cross approximation
8. Preconditioning
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 46 / 183
Review of classical BEM concepts
InterpolationI Partition ∂Ω into elements (possibly curvilinear and with curvilinear edges):
∂Ω = ∪Nee=1Ee
I Isoparametric representation (most commonly used) of ∂Ω and unknown φ:
x =
n(e)∑q=1
Nq(a)xq
φ(x) =
n(e)∑q=1
Nq(a)φq
(a = (a1, a2) ∈ ∆e)
ξ( )
a
∆
1
2
3
12
a
e
E
_
a
_
a
ex
xx
I Connectivity table:
Q(e, q) global number of q-th node of Ee (1≤ e ≤Ne , 1≤ q≤ n(e))
I Isoparametric interpolation: N = NN (for scalar problems).
(with NN: number of nodes and N: number of unknown nodal DOFs)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 47 / 183
Review of classical BEM concepts
Collocation BEMSample integral equation (Laplace + Dirichlet, direct or single-layer formulation):∫
∂Ω
G (x, ξ)ϕ(ξ) dSξ = b(x) (x∈ ∂Ω)
I Principle: enforce integral equation at the NN nodes x = x1, . . . , xNN .I Leads to linear system of equations
Aϕ = b (A∈RN×N , b∈RN)
where
APQ =∑
e∈I (Q)
∫∆e
G (xP , ξ(a))NQ(a)J(a)da (1 ≤ P,Q ≤ N)
bP = b(xP)
I Matrix A square, fully-populated, invertible, non-symmetric obtained byassembly of element matrices
Ae(xP) ∈ R1,n(e) =
[∫∆e
G (xP , ξ(a))Nq(a)J(a)da
]1≤q≤n(e)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 48 / 183
Review of classical BEM concepts
Evaluation of element integrals
Ae(xP) ∈ R1,n(e) =
[∫∆e
G (xP , ξ(a))Nq(a)J(a)da
]1≤q≤n(e)
I If xP 6∈Ee (nonsingular element integral): Gaussian quadrature
Ae(xP) ≈G∑
g=1
wgG (xP , ξ(ag ))Nq(ag )J(ag )
I If xP ∈Ee (singular element integral): specialized treatment:I Weakly singular integrals (O(r−1) kernel in 3-D) removed by suitable
transformaation of parametric coordinates aI Strongly singular integrals (O(r−2) in 3-D) either
(i) recast into weakly singular integrals using regularization techniques(ii) evaluated directly as Cauchy principal values
I For simple element shapes and interpolations (e.g. 3-noded isoparametrictriangle), analytic singular integration available
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 49 / 183
Review of classical BEM concepts
Galerkin BEMExample (simplest): solve Dirichlet problem for Laplace equation using single-layerpotential
I Matrix A square, fully-populated, invertible, symmetric
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 50 / 183
Review of classical BEM concepts
Limitations of “traditional” BEM
CPU for the main steps of traditional BEMs:
(a) Set-up of A: CPU = O(N2);
(b) Solution using direct solver (usually LU factorization): CPU = (N3);
(c) Evaluation of integral representations at M points: CPU = O(N ×M).
Besides:
(d) O(N2) memory needed for storing A.=⇒ Problem size N at most O(104)
Reasons fors (a)-(d):
I G (x, ξ) non-zero for all (x, ξ);
I Element matrices Ae(xP) recomputed for each new collocation point xP .
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 51 / 183
Review of classical BEM concepts
Overcoming the limitations of “traditional” BEM
Two issues:
1. To accelerate the BEM (i.e. to reduce its O(N3) complexity)
2. To increase permitted problem sizes.
Main ideas:
(i) Iterative solution of BEM matrix equation=⇒ CPU = O(N2 × NI), with usually NI/N → 0;
(ii) Acceleration of matrix-vector product Aϕ for given density ϕ.=⇒ complexity lower than O(N2).
Several strategies available for developing fast BEMsThe Fast Multipole Method (FMM) is the most developed to date.
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 52 / 183
The GMRES iterative solver
1. Review of boundary integral equation formulationsElectrostaticsLaplaceElastostaticsFrequency-domain wave equations
2. Review of classical BEM concepts3. The GMRES iterative solver4. The fast multipole method (FMM) for the Laplace equation
Multipole expansion of 1/rThe single-level fast multipole methodThe multi-level fast multipole method
5. The fast multipole method (FMM) for elastostatics6. The fast multipole method for elastodynamics7. Other acceleration methods
Exponential representation of 1/rFMM using equivalent sourcesClustering and low-rank approximationsKernel-independent acceleration via kernel interpolationAdaptive cross approximation
8. Preconditioning
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 53 / 183
The GMRES iterative solver
The GMRES algorithm
I Linear system
Au = b
A∈RN×N or CN×N , A invertibleu∈RN ou CN ,b∈RN ou CN
I Generalized Minimal RESiduals (GMRES): principle
u(k) = arg minu∈u(0)+Vk
‖b−Au‖2, Vk = Vectv1, . . . , vk to be specified
Explicit form of minimisation:
u(k) = u(0) +k∑
j=1
α(k)j vj
with α(k) ≡ (α(k)1 , . . . , α
(k)k ) = arg min
α1,...,αk
∥∥∥∥r(0) −k∑
j=1
αj Avj
∥∥∥∥2
I Iteration k : basis (v1, . . . , vk−1) augmented with a new vector vk , hence
Vk−1⊂Vk .
Minimization problem size increases with k : restart when k >m, GMRES(m)Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 54 / 183
The GMRES iterative solver
The GMRES algorithm
I If k = N, one must have u(N) = u (hence convergence within ≤ N iterationsI In practice: (i) Niter N is desired, (ii) exact convergence not necessary.
r (k) ≡ ‖b− Au(k)‖ ≤ εr (0) (ε: tolerance)
I Construction of subspace Vk = Vectv1, . . . , vk using Krylov vectors:
I Sequence (v1, . . . , vk) constructed using orthonormalization of Krylovvectors (w1, . . . ,wk):
vT
`vk = 0
‖vk‖ = 1
Vectw1, . . . ,wk = Vectv1, . . . , vk = Vk
for all k ≥ 1
Main contribution to computational cost: evaluation of matrix-vectorproducts Aw for given w.
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 55 / 183
The GMRES iterative solver
PreconditioningI Left preconditioning:
M−1Au = M−1b
Improved convergence (i.e. less iterations) if M−1A better conditioned thanA
I Krylov sequence associated with matrix M−1A:
r(0) = M−1b−M−1Au(0) = w1 i.e. Mw1 = b− Au(0)
wk+1 = M−1Awk i.e. Mwk+1 = Awk (k ≥ 0)
I Modified convergence criterion:
‖M−1b−M−1Au(k)‖ ≤ ε‖M−1b‖
I Many approaches available for definir preconditioning matrices M:→ Diagonal preconditioneur Mij = Aijδij ;→ Incomplete LU factorization of A;→ Sparse approximate inverses;→ Multigrid approaches;→ Preconditioners exploiting specific features of the problems, e.g.
single-inclusion case for many-inclusion problems.
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 56 / 183
The fast multipole method (FMM) for the Laplace equation
1. Review of boundary integral equation formulationsElectrostaticsLaplaceElastostaticsFrequency-domain wave equations
2. Review of classical BEM concepts3. The GMRES iterative solver4. The fast multipole method (FMM) for the Laplace equation
Multipole expansion of 1/rThe single-level fast multipole methodThe multi-level fast multipole method
5. The fast multipole method (FMM) for elastostatics6. The fast multipole method for elastodynamics7. Other acceleration methods
Exponential representation of 1/rFMM using equivalent sourcesClustering and low-rank approximationsKernel-independent acceleration via kernel interpolationAdaptive cross approximation
8. Preconditioning
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 57 / 183
The fast multipole method (FMM) for the Laplace equation
Origins of the FMM: fast computation of potentials
Φ(xi ) = C
Nξ∑j=1
qj
‖ξj −xi‖(1≤ i ≤Nx)
C = (4πε0)−1, electric charges qj (electrostatic); C =G, masses qj (gravitation)
I Straightforward computation: CPU = O(NxNξ);I Reason: influence coefficient ‖ξj −xi‖−1 depends on both xi and ξj ;I Fast summation (Greengard, 1985): CPU = O(Nx + Nξ)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 58 / 183
The fast multipole method (FMM) for the Laplace equation
Iterative solution of integral equation
Model problem:
find ϕ,
∫∂Ω
G (x, ξ)ϕ(ξ) dSξ = b(x), i.e. S[ϕ, ∂Ω](x) = b(x) (x ∈ ∂Ω)
Krylov vector: Aϕ discretized version of
S[ϕ, ∂Ω](x) :=
∫∂Ω
G (x, ξ)ϕ(ξ) dSξ
Integral operator S: a generalization to infinite-dimensional function spaces (hereH−1/2(∂Ω)) of the concept of matrix.
I Using traditional BEM: CPU = O(N2) for each evaluation of Aϕ;
I Aim of the Fast Multipole Method: evaluation of Aϕ at CPU cost lowerthan O(N2).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 59 / 183
The fast multipole method (FMM) for the Laplace equation
FMM: main ideas
S[ϕ, ∂Ω](x) :=
∫∂Ω
G (x, ξ)ϕ(ξ) dSξ
I Main idea: seek to reuse element integrations (w.r.t. ξ) when collocationpoint x is changed;
I Method: express the fundamental solution as a series of products:
G (x, ξ) =∞∑n=0
gn(x)hn(ξ)
and truncate the series at suitable level p:
G (x, ξ) =
p∑n=0
gn(x)hn(ξ) + εG (p)
I Consequence:
S[ϕ, ∂Ω](x) =
p∑n=0
gn(x)
∫∂Ω
hn(ξ)ϕ(ξ) dSξ + ε(p)
The p integrations are independent on x and are reusable as x is changed.Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 60 / 183
The fast multipole method (FMM) for the Laplace equation
FMM: main ideas
How to express G (x, ξ) as a sum (series) of products? Taylor expansionabout origins x0 and ξ0:
G (x, ξ) =∑m≥0
1
m![∂mξ G ](x, ξ0) (ξ−ξ0)m
=∑n≥0
∑m≥0
1
m!n![∂nx∂
mξ G ](x0, ξ0) (x−x0)n(ξ−ξ0)m
ξ
ξ0 x0
x
r r0
For Laplace kernel 1/r : sophisticated version of Taylor expansion leads tomultipole expansion (see next).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 61 / 183
The fast multipole method (FMM) for the Laplace equation Multipole expansion of 1/r
1. Review of boundary integral equation formulations
2. Review of classical BEM concepts
3. The GMRES iterative solver
4. The fast multipole method (FMM) for the Laplace equationMultipole expansion of 1/rThe single-level fast multipole methodThe multi-level fast multipole method
5. The fast multipole method (FMM) for elastostatics
6. The fast multipole method for elastodynamics
7. Other acceleration methods
8. Preconditioning
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 62 / 183
The fast multipole method (FMM) for the Laplace equation Multipole expansion of 1/r
Multipole expansion of 1/rThe Multipole expansion of 1/r is given (see proof later) by:
1
‖ξ−x‖ =+∞∑n=0
n∑m=−n
Rn,m(x−x0)+∞∑n′=0
n′∑m′=−n′
(−1)nSn+n′,m+m′(ξ0−x0)Rn′,m′(ξ−ξ0)
Rn,m(z) =1
(n + m)!Pmn (cosα)e imβρn
Sn,m(z) = (n −m)!Pmn (cosα)e imβ 1
ρn+1
with z = ρ[sinα(cosβ e1 + sinβ e2)
)+ cosα e3
]Conditions for convergence of the multipole expansion:
‖x−x0‖ < ‖ξ − x0‖ and ‖ξ−ξ0‖ < ‖x−ξ0‖
ξ
ξ0 x0
x
r r0
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 63 / 183
The fast multipole method (FMM) for the Laplace equation Multipole expansion of 1/r
Multipole expansion of 1/r : computation of solid harmonics Rn,m,Sn,m
Solid harmonics Rn,m and Sn,m evaluated using Cartesian coordinates usingrecursion formulae derived from those for Legendre polynomials:
I The Rn,m(z) are computed recursively by setting R0,0(z) = 1 and using
I Finally, Rn,m(z) and Sn,m(z) for negative values of m are computed via theidentities
Rn,−m(z) = (−1)mRn,m(z) Sn,−m(z) = (−1)mSn,m(z)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 64 / 183
The fast multipole method (FMM) for the Laplace equation Multipole expansion of 1/r
Multipole expansion of 1/r : computation of solid harmonics Rn,m,Sn,m
Derivatives of the Rn,m:
∂
∂z1Rn,m(z) =
1
2(Rn−1,m−1 − Rn−1,m+1)(z)
∂
∂z2Rn,m(z) =
i
2(Rn−1,m−1 + Rn−1,m+1)(z)
∂
∂z3Rn,m(z) = Rn−1,m
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 65 / 183
The fast multipole method (FMM) for the Laplace equation Multipole expansion of 1/r
Multipole expansion of 1/r : proof (1/4)
ξ
ξ0
xr
G (x, ξ) =1
4π‖ξ−x‖ξ − x = (ξ−ξ0)− (x−ξ0)
Spherical coordinates:
(ξ−ξ0) = ρ[sinα(cosβ e1 + sinβ e2) + cosα e3
](x−ξ0) = R
[sin θ(cosϕ e1 + sinϕ e2) + cos θ e3
]1
‖ξ−x‖ = (R2 − 2ρR cos Φ + ρ2)−1/2
cos Φ =1
ρR(ξ−ξ0)·(x−ξ0) = sinα sin θ cos(β − ϕ) + cosα cos θ
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 66 / 183
The fast multipole method (FMM) for the Laplace equation Multipole expansion of 1/r
Multipole expansion of 1/r : proof (2/4)
1
‖ξ−x‖ = (R2 − 2ρR cos Φ + ρ2)−1/2
=1
R(1− 2zt + t2)−1/2
(z = cos Φ, t =
ρ
R
)Since (1−2zt + t2)−1/2 is the generating function of the Legendre polynomials,i.e.:
(1− 2zt + t2)−1/2 =∑n≥0
Pn(z)tn (t < 1)
one has:
1
‖ξ−x‖ =+∞∑n=0
Pn(cos Φ)
Rn+1ρn (ρ < R)
ξ
ξ0
xr
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 67 / 183
The fast multipole method (FMM) for the Laplace equation Multipole expansion of 1/r
Multipole expansion of 1/r : proof (3/4)
1
‖ξ−x‖ =+∞∑n=0
Pn(cos Φ)
Rn+1ρn (ρ < R)
To recast as a series of products g(ρ, α, β)h(R, θ, ϕ): addition theorem forLegendre polynomials:
Pn(cos Φ) =n∑
m=−n
(n −m)!
(n + m)!
(Pmn (cosα)e imβ
)(Pmn (cos θ)e−imϕ
)1
‖ξ−x‖ =+∞∑n=0
n∑m=−n
Rn,m(ξ−ξ0)Sn,m(x−ξ0)
Rn,m(z) =1
(n + m)!Pmn (cosα)e imβρn
Sn,m(z) = (n −m)!Pmn (cos θ)e imϕ 1
Rn+1
The series is convergent if ‖ξ−ξ0‖ < ‖x− ξ0‖.Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 68 / 183
The fast multipole method (FMM) for the Laplace equation Multipole expansion of 1/r
Multipole expansion of 1/r : proof (4/4)
1
‖ξ−x‖ =+∞∑n=0
n∑m=−n
Rn,m(ξ−ξ0)Sn,m(x−ξ0)
Application to evaluation of potentials:
Φ(xi ) = C
Nξ∑j=1
qj
‖ξj − xi‖= C
+∞∑n=0
n∑m=−n
Mn,m(ξ0)Sn,m(xi −ξ0)
with multipole moments defined by
Mn,m(ξ0) =
Nξ∑j=1
qjRn,m(ξj −ξ0)
Truncation of series to n< p (with error control available, see later):
I Evaluation of NξNx influence coefficients ‖ξj −xi‖−1 replaced with
evaluation of p2Nx products Mn,m(ξ0)Sn,m(xi −ξ0)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 69 / 183
The fast multipole method (FMM) for the Laplace equation Multipole expansion of 1/r
Multipole expansion of 1/r : proofInsertion of local origin x0 into xi −ξ0:
I Note identity
Sn,m(z) = (−1)n( ∂
∂z1+ i
∂
∂z2
)m ∂n−m
∂zn−m3
1
‖z‖ (m > 0) (a)
= (−1)n+m( ∂
∂z1− i
∂
∂z2
)−m ∂n+m
∂zn+m3
1
‖z‖ (m < 0) (b)
I Invoke multipole expansion of 1/‖z‖ with z = (xi −x0)− (ξ0−x0):
1
‖xi −ξ0‖=
+∞∑n′=0
n∑m′=−n′
Rn′,m′(xi −x0)Sn′,m′(ξ0−x0)
I Use representation (a,b) for Sn′,m′(ξ0−x0);I Exploit harmonicity of 1/‖ξ0−x0‖ via( ∂
∂ξ01
+ i∂
∂ξ02
)( ∂
∂ξ01
− i∂
∂ξ02
) 1
‖ξ0−x0‖+
∂2
∂ξ03
2
1
‖ξ0−x0‖= 0
I Reorder and reorganize resulting formula
Sn,m(xi −ξ0) =+∞∑n′=0
n∑m′=−n′
Rn′,m′(xi −x0)Sn+n′,m+m′(ξ0−x0)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 70 / 183
The fast multipole method (FMM) for the Laplace equation Multipole expansion of 1/r
Multipole expansion: truncation error
Assume one can find R > 0 and χ > 1 such that(‖x−x0‖ < R et ‖ξ − x0‖ > χR
)et
(‖ξ−ξ0‖ < R et ‖x−ξ0‖ > χR
)An upper bound of the error arising from truncating the multipole expansion atorder p is:∣∣∣∣ 1
‖ξ−x‖ −p∑
n=0
n∑m=−n
Rn,m(x−x0)
p∑n′=0
n′∑m′=−n′
(−1)nSn+n′,m+m′(ξ0−x0)Rn′,m′(ξ−ξ0)
∣∣∣∣ ≤ 1
R(χ− 1)χp+1
ξ
ξ0 x0
x
r r0
The truncation error is scale-independentMarc Bonnet (POems, ENSTA) Methode d’elements de frontiere 71 / 183
The fast multipole method (FMM) for the Laplace equation The single-level fast multipole method
1. Review of boundary integral equation formulations
2. Review of classical BEM concepts
3. The GMRES iterative solver
4. The fast multipole method (FMM) for the Laplace equationMultipole expansion of 1/rThe single-level fast multipole methodThe multi-level fast multipole method
5. The fast multipole method (FMM) for elastostatics
6. The fast multipole method for elastodynamics
7. Other acceleration methods
8. Preconditioning
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 72 / 183
The fast multipole method (FMM) for the Laplace equation The single-level fast multipole method
Single-level fast multipole methodBoundary of interest enclosed ina cubic grid d
∂Ω
Convergence of multipoleexpansion guaranteed if x and ξlie in non-adjacent cells, with
χ ≥√
3
Cx
Cξ ∈ (A(Cx))Cξ 6∈ (A(Cx))
Ω
d
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 73 / 183
The fast multipole method (FMM) for the Laplace equation The single-level fast multipole method
Single-level FMM
I Matrix-vector product ←− integral operator evaluation
S[ϕ, ∂Ω](x) =
∫∂Ω
G (x, ξ)ϕ(ξ) dSξ
I Split into near and far contributions:∫∂Ω
=∑
Cξ∈A(Cx )
∫∂Ω∩Cξ
+∑
Cξ /∈A(Cx )
∫∂Ω∩Cξ
S[ϕ, ∂Ω](x) = S[ϕ, ∂Ω]near(x) + S[ϕ, ∂Ω]far(x)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 74 / 183
The fast multipole method (FMM) for the Laplace equation The single-level fast multipole method
Single-level FMMFar contribution S[ϕ, ∂Ω]far(x):
S[ϕ, ∂Ω]far(x) =∑
Cξ /∈A(Cx )
∫∂Ω∩Cξ
G (x, ξ)ϕ(ξ) dSξ (x∈Cx)
Introduce (truncated) multipole expansion of G (x, ξ), with ξ0 and x0 chosen ascentres of cells Cξ and Cx :
The fast multipole method (FMM) for the Laplace equation The single-level fast multipole method
Single-level FMM: optimal complexity
Total CPU time for one evaluation of S[ϕ, ∂Ω]:
CPU = Ap2N + NB
(Bp4NB + Cp2(N/NB) + D |A(Cx)|N2/N2
B
)= (A + C )p2N + Bp4N2
B + DN2/NB
Optimal choice: NB = O(N2/3), yielding CPU / GMRES iteration = O(N4/3)
I Single-level FMM (Laplace and other elliptic PDEs): CPU = O(N4/3);
I To further improve complexity: multi-level FMM
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 78 / 183
The fast multipole method (FMM) for the Laplace equation The multi-level fast multipole method
1. Review of boundary integral equation formulations
2. Review of classical BEM concepts
3. The GMRES iterative solver
4. The fast multipole method (FMM) for the Laplace equationMultipole expansion of 1/rThe single-level fast multipole methodThe multi-level fast multipole method
5. The fast multipole method (FMM) for elastostatics
6. The fast multipole method for elastodynamics
7. Other acceleration methods
8. Preconditioning
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 79 / 183
The fast multipole method (FMM) for the Laplace equation The multi-level fast multipole method
Multi-level FMM1
1
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 80 / 183
The fast multipole method (FMM) for the Laplace equation The multi-level fast multipole method
Multi-level FMM: initialization of multipole moments
(v) Evaluation of local expansions at leaf cells:CPU = O(p2N/M)
(vi) Evaluation of near contributions S[ϕ, ∂Ω]near(x) using standard BEMtechniques:
CPU = O((N/M)×M × |A(C)|M) = O(|A(Cx)|MN)
Overall complexity: CPU = O(N)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 91 / 183
The fast multipole method (FMM) for elastostatics
1. Review of boundary integral equation formulationsElectrostaticsLaplaceElastostaticsFrequency-domain wave equations
2. Review of classical BEM concepts3. The GMRES iterative solver4. The fast multipole method (FMM) for the Laplace equation
Multipole expansion of 1/rThe single-level fast multipole methodThe multi-level fast multipole method
5. The fast multipole method (FMM) for elastostatics6. The fast multipole method for elastodynamics7. Other acceleration methods
Exponential representation of 1/rFMM using equivalent sourcesClustering and low-rank approximationsKernel-independent acceleration via kernel interpolationAdaptive cross approximation
8. Preconditioning
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 92 / 183
The fast multipole method (FMM) for elastostatics
Elastostatics
Reformulation of Kelvin solution in terms of 1/r :
Uki (x, ξ) =
1
16πµ(1−ν)
(3−4ν)δik
1
r+ (ξi − xi )
∂
∂xk
1
r
(a)
T ki (x, ξ) = − 1
8πµ(1−ν)
(1−2ν)
[nk(ξ)
∂
∂xi
1
r− ni (ξ)
∂
∂xk
1
r
]+ 2(1−ν)δiknj(ξ)
∂
∂xj
1
r+ (ξj − xj)nj(ξ)
∂2
∂xi∂xk
1
r
(b)
Substitute multipole expansion of 1/r into (a) and (b):
1
‖ξ−x‖ =+∞∑n=0
n∑m=−n
Rn,m(x−x0)+∞∑n′=0
n′∑m′=−n′
(−1)nSn+n′,m+m′(ξ0−x0)Rn′,m′(ξ−ξ0)
(again)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 93 / 183
The fast multipole method (FMM) for elastostatics
Elastostatics: multipole expansion of Kelvin solution
Uki (x, ξ) =
1
16πµ(1− ν)
+∞∑n′=0
n′∑m′=−n′
F n′,m′
ki (x−ξ0)Rn′,m′(ξ−ξ0)
+ G n′,m′
k (x−ξ0)(ξi −ξi0)Rn′,m′(ξ−ξ0)
F n′,m′
ki (x−ξ0) =+∞∑n=0
n∑m=−n
(−1)nSn+n′,m+m′(ξ0−x0)
[(3− 4ν)δikRn,m(x−x0) + (ξi0−xi0 − xi −xi0)
∂
∂xkRn,m(x−x0)
]G n′,m′
k (x−ξ0) =+∞∑n=0
n∑m=−n
(−1)nSn+n′,m+m′(ξ0−x0)∂
∂xkRn,m(x−x0)
A similar formula (not shown) can be established for the multipole representationof T k
i (x, ξ)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 94 / 183
The fast multipole method (FMM) for elastostatics
Elastostatics: multipole moments
M tn′,m′;i (Cξ; ξ0) =
∫∂Ω∪Cξ
Rn′,m′(ξ−ξ0) ti (ξ) dSξ
M tn′,m′(Cξ; ξ0) =
∫∂Ω∪Cξ
Rn′,m′(ξ−ξ0) (ξi −ξi0)ti (ξ) dSξ
Mun′,m′;ki (Cξ; ξ0) =
∫∂Ω∪Cξ
Rn′,m′(ξ−ξ0) nk(ξ)ui (ξ) dSξ
Mun′,m′;k(Cξ; ξ0) =
∫∂Ω∪Cξ
Rn′,m′(ξ−ξ0) nk(ξ)(ξi −ξi0)ui (ξ) dSξ
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 95 / 183
The fast multipole method (FMM) for elastostatics
M2M, M2L and L2L formulaeM2M, M2L and L2L formulae are derived using those for 1/r . For example, theelastostatic M2M formulae are:
M tn,m;i (C(`)
ξ ; ξ(`−1)0 ) =
∞∑n′=0
n′∑m′=−n′
Rn′,m′(ξ(`)0 −ξ
(`−1)0 )M t
n−n′,m−m′;i (C(`)ξ ; ξ
(`)0 )
M tn,m(C(`)
ξ ; ξ(`−1)0 ) =
∞∑n′=0
n′∑m′=−n′
Rn′,m′(ξ
(`)0 −ξ
(`−1)0 )Mn−n′,m−m′(C(`)
ξ ; ξ(`)0 )
+ (ξ(`)i0 − ξ
(`−1)i0 )M t
n−n′,m−m′;i (C(`)ξ ; ξ
(`)0 )
Mun,m;ki (C(`)
ξ ; ξ(`−1)0 ) =
∞∑n′=0
n′∑m′=−n′
Rn′,m′(ξ(`)0 −ξ
(`−1)0 )Mn−n′,m−m′;ki (C(`)
ξ ; ξ(`)0 )
Mun,m;k(C(`)
ξ ; ξ(`−1)0 ) =
∞∑n′=0
n′∑m′=−n′
Rn′,m′(ξ
(`)0 −ξ
(`−1)0 )Mn−n′,m−m′;k(C(`)
ξ ; ξ(`)0 )
+ (ξ(`)i0 − ξ
(`−1)i0 )M t
n−n′,m−m′;ki (C(`)ξ ; ξ
(`)0 )
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 96 / 183
The fast multipole method (FMM) for elastostatics
Numerical example: uniform thermal strain in ellipsoidal region
x
yz
a1
a2
a3
FEM
BEM
e
x
z
y
FEM
BEM
Mesh Nodes Elements Oct-tree DOFs
BEM FEM Max level Leaves NBEM NF NBEM + NF
1 267 346 979 3 42 1050 276 1326
2 822 1038 3153 3 100 3126 903 4029
3 1362 1540 5563 3 103 4632 1770 6402
4 2274 2418 9626 4 301 7266 3189 10455
5 5881 5200 26602 4 422 15612 9837 25449
6 12868 9402 61770 5 1175 28218 24495 52713
7 20258 12842 100200 6 1403 38538 41505 80043
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 97 / 183
The fast multipole method (FMM) for elastostatics
Numerical example: uniform thermal strain in ellipsoidal region
-0.096
-0.095
-0.094
-0.093
-0.092
-0.091
0 1 2 3 4 5 6 7 8 9 10
s
Sxx
Exact
Mesh 3
Mesh 5
Mesh 7(a)
-0.077
-0.076
-0.075
-0.074
-0.073
-0.072
0 1 2 3 4 5 6 7 8 9 10
s
Syy
Exact
Mesh 3
Mesh 5
Mesh 7
(b)
-0.057
-0.056
-0.055
-0.054
0 1 2 3 4 5 6 7 8 9 10
s
Szz
Exact
Mesh 3
Mesh 5
Mesh 7
(c)
Mesh Precond. (s) FMM Iters Total time
BEM FEM (s/iter) n (s)
1 10 <1 1 37 43
2 36 <1 2 37 154
3 50 <1 3 37 202
4 64 3 6 36 277
5 169 18 11 37 721
6 349 101 19 38 1425
7 512 279 42 38 2913
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+03 1.E+04 1.E+05
DOFs
Total Time (sec.)
FMM-CBEM
Traditional CBEM
Cross over point
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 98 / 183
The fast multipole method (FMM) for elastostatics
Numerical example: many-inclusion problem
Nodes Elements Oct-tree DOFs
BEM FEM Max. level Leaves NBEM NF NBEM + NF
93227 122880 326493 5 7176 374784 92289 467073
Precond. (s) Time (s) Iters Total time
BEM FEM Upw. Downw. Direct Cycle n (s)
6609 19 47 48 84 180 147 39656
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 99 / 183
The fast multipole method (FMM) for elastostatics
Numerical example: Pian Telessio dam
Dam
Rock
Water
10501010
448
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 100 / 183
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 101 / 183
The fast multipole method (FMM) for elastostatics
Numerical example: Pian Telessio dam
-4.00E-03
-3.00E-03
-2.00E-03
-1.00E-03
0.00E+00
1.00E-03
2.00E-03
3.00E-03
4.00E-03
5.00E-03
0 100 200 300 400 500
s (m)
Ux (m)
ABAQUS
Mesh 1
Mesh 2
Mesh 3
(a)
-1.60E-02
-1.40E-02
-1.20E-02
-1.00E-02
-8.00E-03
-6.00E-03
-4.00E-03
-2.00E-03
0.00E+00
0 100 200 300 400 500
s (m)
Uy (m)
ABAQUS
Mesh 1
Mesh 2
Mesh 3
y
z
x
s
(b)
-1.80E-03
-1.60E-03
-1.40E-03
-1.20E-03
-1.00E-03
-8.00E-04
-6.00E-04
-4.00E-04
-2.00E-04
0 100 200 300 400 500
s (m)
Uz (m)
ABAQUS
Mesh 1
Mesh 2
Mesh 3
(c)
-4.0E+05
-3.5E+05
-3.0E+05
-2.5E+05
-2.0E+05
-1.5E+05
-1.0E+05
0 5 10 15 20 25 30
s (m)
Sxx (N/m2)
ABAQUS
Mesh 1
Mesh 2
Mesh 3
-1.4E+06
-1.2E+06
-1.0E+06
-8.0E+05
-6.0E+05
-4.0E+05
-2.0E+05
0.0E+00
2.0E+05
0 5 10 15 20 25 30
s (m)
Syy (N/m2)
ABAQUS
Mesh 1
Mesh 2
Mesh 3
a=151 m
s
x
y
z
-2.0E+06
-1.6E+06
-1.2E+06
-8.0E+05
-4.0E+05
0.0E+00
4.0E+05
8.0E+05
0 5 10 15 20 25 30
s (m)
Szz (N/m2)
ABAQUS
Mesh 1
Mesh 2
Mesh 3
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 102 / 183
The fast multipole method (FMM) for elastostatics
Exemple, calcul d’amortissements dans les MEMS
Figure 5. More realistic model of one-fourth of the MEMS: geometry and mesh.
Frangi A., Spinola G., Vigna B., IJNME 68:1031–1051Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 103 / 183
The fast multipole method (FMM) for elastostatics
Exemple, calcul d’amortissements dans les MEMSTable II. Comparison between meshes of increasing refinement.
Mesh employed Mesh 1 Mesh 2 Mesh 3
Number of unknowns 125 058 272 364 548 388
Global force (N) 1.80 × 10−4 2.01 × 10−4 2.12 × 10−4
Figure 6. Convergence of the GMRES solver and of the force computed.
Frangi A., Spinola G., Vigna B., IJNME 68:1031–1051Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 104 / 183
The fast multipole method (FMM) for elastostatics
Exemple, homogeneisation numerique
Fig. 10 Contour plot of surface stresses „Ãs`… for a model with 216 ‘‘randomly’’ distributed
and oriented short fibers
Fig. 9 A BEM mesh used for the short fiber inclusion „with 456elements…
Liu Y.J., Nishimura N. et al., ASME J. Appl. Mech. 72:115–128 (2005)Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 105 / 183
The fast multipole method (FMM) for elastostatics
Exemple, homogeneisation numerique
Fig. 11 A RVE containing 2197 short fibers with the total DOFÄ3 018 678
Liu Y.J., Nishimura N. et al., ASME J. Appl. Mech. 72:115–128 (2005)Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 106 / 183
The fast multipole method (FMM) for elastostatics
Exemple, homogeneisation numerique
Fig. 14 A RVE containing 5832 long fibers with the total DOFÄ10 532 592
Liu Y.J., Nishimura N. et al., ASME J. Appl. Mech. 72:115–128 (2005)Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 107 / 183
The fast multipole method (FMM) for elastostatics
Exemple, homogeneisation numerique
Fig. 15 Estimated effective Young’s moduli in the x-direction for the composite model withup to 5832 long rigid fibers „fiber volume fractionÄ3.85%…
Liu Y.J., Nishimura N. et al., ASME J. Appl. Mech. 72:115–128 (2005)Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 108 / 183
The fast multipole method for elastodynamics
1. Review of boundary integral equation formulationsElectrostaticsLaplaceElastostaticsFrequency-domain wave equations
2. Review of classical BEM concepts3. The GMRES iterative solver4. The fast multipole method (FMM) for the Laplace equation
Multipole expansion of 1/rThe single-level fast multipole methodThe multi-level fast multipole method
5. The fast multipole method (FMM) for elastostatics6. The fast multipole method for elastodynamics7. Other acceleration methods
Exponential representation of 1/rFMM using equivalent sourcesClustering and low-rank approximationsKernel-independent acceleration via kernel interpolationAdaptive cross approximation
8. Preconditioning
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 109 / 183
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 133 / 183
The fast multipole method for elastodynamics
Example: scattering of a plane P wave by a semi-elliptical canyon
B
C
Dx
y
z
a
b
a
y
zplane P wave
a
D = 6a
free surface
elastic half-space
D EBA
C
θ0
I Simplified configuration for topographic site effect
I Low frequency: comparison with other published results
I Higher frequency: FMM
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 134 / 183
The fast multipole method for elastodynamics
Example: scattering of a plane P wave by a semi-elliptical canyon
Comparaison with earlier results, kSa = 0.5 (low frequency), N = 25 788
-3 -2 -1 0 1 2 3s / R
0.5
1
1.5
2
2.5
3
3.5di
spla
cem
ent m
odul
us
|uy| (present FMM)
|uy| (Reinoso et al.)
|uz| (present FMM)
|uz| (Reinoso et al.)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 135 / 183
The fast multipole method for elastodynamics
Example: scattering of a plane P wave by a semi-elliptical canyon
Results for higher frequency ksa = 2
0.012
0.630
1.250
1.870
2.500
x
y0.28
0.97
1.66
2.35
3.05
N = 353 232 (32 iter., 140 s / iter, single-proc. 3 GHz PC)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 136 / 183
The fast multipole method for elastodynamics
Example: scattering of a plane P wave by an alluvial hemisphericvalley
x, y
z
plane P wave
R
D = 5R
free surface
elastic half-spaceA
B C
(N = 17409)
I µ2 = 0.3µ1, ρ2 = 0.6ρ1, ν1 = 0.25, ν2 = 0.3
I Low frequency: comparison with Sanchez-Sesma (1983) and Delavaud (2007)
I Higher frequency: FMM
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 137 / 183
The fast multipole method for elastodynamics
Example: scattering of a plane P wave by an alluvial hemisphericvalley
Comparaison with earlier results, kPR = 0.5
I Sanchez-Sesma, 1983 (semi-analytical)
I Delavaud, 2007 (spectral finite element method)
x, yz
R
D = 5R
A
B Cs
0 1 2x/R
0
1
2
3
4
5
6
disp
lace
men
t mod
ulus
|uy| (present FMM)
|uy| (Sanchez-Sesma 1983)
|uy| (Delavaud 2007, SEM)
|uz| (present FMM)
|uz| (Sanchez-Sesma 1983)
|uz| (Delavaud 2007, SEM)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 138 / 183
The fast multipole method for elastodynamics
Example: scattering of a plane P wave by an alluvial hemisphericvalley
Comparaison with earlier results, kPR = 0.7:
I Sanchez-Sesma, 1983 (semi-analytical)
I Delavaud, 2007 (spectral finite element method)
x, yz
R
D = 5R
A
B Cs
0 2x/R
0
1
2
3
4
5
6
disp
lace
men
t mod
ulus
|uy| (present FMM)
|uy| (Sanchez-Sesma 1983)
|uy| (Delavaud 2007, SEM)
|uz| (present FMM)
|uz| (Sanchez-Sesma 1983)
|uz| (Delavaud 2007, SEM)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 139 / 183
The fast multipole method for elastodynamics
Example: scattering of a plane P wave by an alluvial hemisphericvalley
Results for a higher frequency kPR = 1N = 84 882 (76 iterations, single-proc. 3 GHz PC)
x, yz
R
D = 5R
A
B Cs
0 1 2x/R
0
1
2
3
4
5
6
disp
lace
men
t mod
ulus
|uy| (present FMM)
|uz| (present FMM)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 140 / 183
Other acceleration methods
1. Review of boundary integral equation formulationsElectrostaticsLaplaceElastostaticsFrequency-domain wave equations
2. Review of classical BEM concepts3. The GMRES iterative solver4. The fast multipole method (FMM) for the Laplace equation
Multipole expansion of 1/rThe single-level fast multipole methodThe multi-level fast multipole method
5. The fast multipole method (FMM) for elastostatics6. The fast multipole method for elastodynamics7. Other acceleration methods
Exponential representation of 1/rFMM using equivalent sourcesClustering and low-rank approximationsKernel-independent acceleration via kernel interpolationAdaptive cross approximation
8. Preconditioning
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 141 / 183
Other acceleration methods Exponential representation of 1/r
1. Review of boundary integral equation formulations
2. Review of classical BEM concepts
3. The GMRES iterative solver
4. The fast multipole method (FMM) for the Laplace equation
5. The fast multipole method (FMM) for elastostatics
6. The fast multipole method for elastodynamics
7. Other acceleration methodsExponential representation of 1/rFMM using equivalent sourcesClustering and low-rank approximationsKernel-independent acceleration via kernel interpolationAdaptive cross approximation
Speeds up M2M, M2L, L2L operations(O(p2N) instead of O(p4N) using ”traditional” FMM)
I Summation w.r.t. (k, i) performed on local expansions, at the very end (afterupward, M2L and downward phases)
I Exponential expansions available for other kernels, e.g. HelmholtzUseful for FMM for low-frequency wave problems
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 146 / 183
Other acceleration methods FMM using equivalent sources
1. Review of boundary integral equation formulations
2. Review of classical BEM concepts
3. The GMRES iterative solver
4. The fast multipole method (FMM) for the Laplace equation
5. The fast multipole method (FMM) for elastostatics
6. The fast multipole method for elastodynamics
7. Other acceleration methodsExponential representation of 1/rFMM using equivalent sourcesClustering and low-rank approximationsKernel-independent acceleration via kernel interpolationAdaptive cross approximation
Other acceleration methods FMM using equivalent sources
FMM using equivalent sourcesMain idea: express fields at remote points in terms of equivalent density, e.g.:∫
∂Ω∩CξG,n(x, ξ)u(ξ) dSξ =
∫Sd
G (x, z)φ(z) dSz for some φ (x 6∈ A(Cξ))
x0
ξ0 Sd
I Ying, L., Biros, G., Zorin, D. A kernel-independent adaptive fastmultipole in two and three dimensions. J. Comp. Phys., 196:591–626 (2004).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 148 / 183
Other acceleration methods FMM using equivalent sources
FMM using equivalent sourcesMain idea: express fields at remote points in terms of equivalent density, e.g.:∫
∂Ω∩CξG,n(x, ξ)u(ξ) dSξ =
∫Sd
G (x, z)φ(z) dSz for some φ (x∈ Sc)
x0
ξ0 Sd
cS
I Solve the above (Fredholm, 1st kind) integral equation for φI Truncation parameter = discretization of φ
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 149 / 183
Other acceleration methods FMM using equivalent sources
FMM using equivalent sources: M2M translations
S(l)d
S(l−1)d
S(l−1)c
Find φ(`−1),
∫S
(`−1)d
G (x, z)φ(`−1)(z) dSz =
∫S
(`)d
G (x, z)φ(`)(z) dSz (x∈ S (`−1)c )
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 150 / 183
Other acceleration methods FMM using equivalent sources
FMM using equivalent sources: L2L translations
(l)dS
S(l−1)d
(l)Sc
Find φ(`),
∫S
(`)d
G (x, z)φ(`)(z) dSz =
∫S
(`−1)d
G (x, z)φ(`−1)(z) dSz (x∈ S (`)c )
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 151 / 183
Other acceleration methods FMM using equivalent sources
FMM using equivalent sources
I Kernel-independent acceleration method;
I Truncation parameter p = discretization of φ (scale-independent for kernelsassociated with elliptic problems);
I Found by Ying, Biros, Zorin (2004) to have O(p2N) complexity / iteration;
I Requires solving 1st kind integral equations (ill-conditioned integral operator)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 152 / 183
Other acceleration methods Clustering and low-rank approximations
1. Review of boundary integral equation formulations
2. Review of classical BEM concepts
3. The GMRES iterative solver
4. The fast multipole method (FMM) for the Laplace equation
5. The fast multipole method (FMM) for elastostatics
6. The fast multipole method for elastodynamics
7. Other acceleration methodsExponential representation of 1/rFMM using equivalent sourcesClustering and low-rank approximationsKernel-independent acceleration via kernel interpolationAdaptive cross approximation
Block clustering + low-rank approximation of blocks = acceleration of matrixoperations
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 157 / 183
Other acceleration methods Clustering and low-rank approximations
SVD and block rank
I Any block A(I , J) of size m×n admits a singular value decomposition(SVD):
A(I , J) = USVT =R∑
k=1
sk ukvT
k s1≥ s2≥ . . . sR > 0, uT
ku` = vT
kv` = δk`
R ≤Min(m, n) is the (numerical) rank of A(I , J)
I A(I , J) has (approximate) low rank r if sk is sufficiently small for k > r
I Computing complete SVD of A(I , J) needs O(mn) memory + O(mn2) CPUnot acceptable; other strategies required−→FMM (analytic decomposition of kernel required)−→Kernel interpolation (analytic decomposition of kernel not required)−→Algebraic treatment of matrix blocks: adaptive cross approximation−→Wavelet transformation of basis functions (not addressed here)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 158 / 183
Other acceleration methods Clustering and low-rank approximations
FMM as block clustering with low-rank approximation of blocks
The multi-level Fast Multipole Method features block clustering (throughhierarchical octree of cubic cells)
and low-rank approximation through truncated multipole expansion
1
‖ξ−x‖ =
p∑n=0
n∑m=−n
Rn,m(x−x0)
p∑n′=0
n′∑m′=−n′
(−1)nSn+n′,m+m′(ξ0−x0)Rn′,m′(ξ−ξ0)
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 159 / 183
Other acceleration methods Kernel-independent acceleration via kernel interpolation
1. Review of boundary integral equation formulations
2. Review of classical BEM concepts
3. The GMRES iterative solver
4. The fast multipole method (FMM) for the Laplace equation
5. The fast multipole method (FMM) for elastostatics
6. The fast multipole method for elastodynamics
7. Other acceleration methodsExponential representation of 1/rFMM using equivalent sourcesClustering and low-rank approximationsKernel-independent acceleration via kernel interpolationAdaptive cross approximation
Other acceleration methods Kernel-independent acceleration via kernel interpolation
Kernel-independent acceleration via kernel interpolation
In many cases:
I Fundamental solution available (not necessary ion closed form);Availability of high-order derivatives problematic at best−→Taylor-based expansion impractical
I Analytic expansion (e.g. multipole, exponential) not available−→FMM treatment impossible
Idea: polynomial interpolation of G (x, ξ) in product of two non-adjacent cells(x∈Cx , ξ∈Cξ)
x0
ξ0
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 161 / 183
Other acceleration methods Kernel-independent acceleration via kernel interpolation
Kernel-independent acceleration via kernel interpolation
G (x, ξ) ≈P∑
p=1
Q∑q=1
Pp(x)G (xp, ξq)Qq(ξ)
xp: interpolation nodes in Cx (Cartesian product of 1-D set of nodes);Pp(x): interpolation polynomials (e.g. Cartesian product of 1-D Lagrange polyn.);
ξq: interpolation nodes in Cξ (Cartesian product of 1-D set of nodes);Qq(x): interpolation polynomials (e.g. Cartesian product of 1-D Lagrange polyn.);
xp
Cξ
Cx
ξq
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 162 / 183
Other acceleration methods Kernel-independent acceleration via kernel interpolation
Kernel-independent acceleration via kernel interpolation
Evaluation of
S[ϕ, ∂Ω]far(x) =∑
Cξ∈A(Cx )
∫∂Ω∩Cξ
G (x, ξ)ϕ(ξ) dSξ
I Multipole moments:
Mq(ξ0) =
∫∂Ω∩Cξ
Qq(ξ)φ(ξ) dSξ
I M2L translation:
Lp(x0) =Q∑
q=1
G (xp, ξq)Mq(ξ0)
I M2M (upward) translations by expressing the Q(`−1)q (ξ; ξ
(`−1)0 ) in terms of
the Q(`)q′ (ξ; ξ
(`)0 ) (e.g. Taylor expansion for polynomials)
I L2L (downward) translations similarly
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 163 / 183
Other acceleration methods Adaptive cross approximation
1. Review of boundary integral equation formulations
2. Review of classical BEM concepts
3. The GMRES iterative solver
4. The fast multipole method (FMM) for the Laplace equation
5. The fast multipole method (FMM) for elastostatics
6. The fast multipole method for elastodynamics
7. Other acceleration methodsExponential representation of 1/rFMM using equivalent sourcesClustering and low-rank approximationsKernel-independent acceleration via kernel interpolationAdaptive cross approximation
Other acceleration methods Adaptive cross approximation
Example: crack propagation analysis
I N ≈ 45000,
I CPU ≈ 4500s
I RAM= 1.5GB
I Kolk, K., Weber, W., Kuhn, G. Investigation of 3D crack propagationproblems via fast BEM formulations. Comp. Mech., 37:32–40 (2005).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 167 / 183
Preconditioning
1. Review of boundary integral equation formulationsElectrostaticsLaplaceElastostaticsFrequency-domain wave equations
2. Review of classical BEM concepts3. The GMRES iterative solver4. The fast multipole method (FMM) for the Laplace equation
Multipole expansion of 1/rThe single-level fast multipole methodThe multi-level fast multipole method
5. The fast multipole method (FMM) for elastostatics6. The fast multipole method for elastodynamics7. Other acceleration methods
Exponential representation of 1/rFMM using equivalent sourcesClustering and low-rank approximationsKernel-independent acceleration via kernel interpolationAdaptive cross approximation
8. Preconditioning
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 168 / 183
Preconditioning
→ Diagonal preconditioneur Mij = Aijδij ;
→ Sparse matrix of near contributions in FMM;
→ Incomplete LU factorization of A;
→ Sparse approximate inverses;
→ Multigrid approaches;
→ Fast BEM solution method (e.g. FMM, ACA) with low truncation;
→ Preconditioners exploiting specific features of the problems, e.g.single-inclusion case for many-inclusion problems.
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 169 / 183
Preconditioning
Sparse approximate inverse (SPAI)
I Definition:
[A]] = arg min[E ]
‖ [I ]− [E ][A] ‖2F [E ] ∈ RN×N sparse
where [A]] has a sparsity pattern (either predefined or found iteratively).
I Hence each column of [A]] solves an uncoupled, small minimization problem:
A]k = arg minE
‖ ek − E[A] ‖ E ∈ R1×N sparse
I Simplification: choose number m of nonzero entries in each row of [A]] andfind SPAI of [A]:
A]i = arg minE∈R1,m
‖ E[Ai ] ‖2 −2 trace(E[Ai ]) + 1
(1≤ i≤N)
where [A] is the sparse matrix made of the m largest entries of [A].
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 170 / 183
Preconditioning
Further reading
I Benzi, M.Preconditioning techniques for large linear systems: a survey.J. Comp. Phys., 182:418–477 (2002).
I Biros, G., Ying, L., Zorin, D.A fast solver for the Stokes equations with distribuuted forces in complexgeometries.J. Comp. Phys., 193:317–348 (2003).
I Bonnet, M.Boundary Integral Equation Method for Solids and Fluids.Wiley (1999).
I Borm, S., Grasedyck, L., Hackbusch, W.Introduction to hierarchical matrices with applications.Engng. Anal. with Bound. Elem., 27:405–422 (2003).
I Carpentieri, B., Duff, I. S., Giraud, L., Sylvand, G.Combining fast mulrtipole techniques and an approximate inverseprecondfitioner for large electromagnetism calculations.SIAM J. Sci. Comput., 27:774–7923 (2005).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 171 / 183
Preconditioning
I Chaillat, S., Bonnet, M., Semblat, J. F.A multi-level fast multipole BEM for 3-D elastodynamics in the frequencydomain.Comp. Meth. Appl. Mech. Engng. (2008, in press).
I Cheng, H., Crutchfield, W. Y., Gimbutas, Z., Greengard, L. F.,Ethridge, J. F., Huang, J., Rokhlin, V., Yarvin, N.A wideband fast multipole method for the Helmholtz equation in threedimensions.J. Comp. Phys., 216:300–325 (2006).
I Chew, W. C. et al.Fast integral equation solvers in computational electromagnetics of complexstructures.Engng. Anal. with Bound. Elem., 27:803–823 (2003).
I Coifman, R., Rokhlin, V., Wandzura, S.The Fast Multipole Method for the Wave Equation : A Pedestrian Prescription.IEEE Antennas and Propagation Magazine, 35:7–12 (1993).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 172 / 183
Preconditioning
I Colton, D., Kress, R.Integral Equation Method in Scattering Theory .John Wiley and sons (1983).
I Darve, E.The fast multipole method: I. Error analysis and asymptotic complexity.SIAM J. Numer. Anal., 38:98–128 (2000).
I Darve, E.The fast multipole method: numerical implementation.J. Comp. Phys., 160:195–240 (2000).
I Darve, E., Have, P.Efficient fast multipole method for low-frequency scattering.J. Comp. Phys., 197:341–363 (2004).
I Dominguez, J.Boundary elements in dynamics.Comp. Mech. Publ., Southampton (1993).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 173 / 183
Preconditioning
I Epton, M.A., Dembart, B.Multipole translation theory for the three-dimensional Laplace and Helmholtzequations.SIAM J. Sci. Comp., 16:865–897 (1995).
I Ergin, A. A., Shanker, B., Michielssen, E.Fast evaluation of three-dimensional transient wave fields using diagonaltranslation operators.J. Comp. Phys., 146:157–180 (1998).
I Fischer, M., Gaul, L.Application of the fast multipole bem for structural-acoustic simulations.Journal of Computational Acoustics, 13:87–98 (2005).
I Fochesato, C., Dias, F.A fast method for nonlinear three-dimensional free-surface waves.Proc. Roy. Soc. A, 462:2715–2735 (2006).
I Fraysse, V., Giraud, L., Gratton, S., Langou, J.A Set of GMRES Routines for Real and Complex Arithmetics on HighPerformance Computers.Tech. rep., CERFACS (2003).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 174 / 183
Preconditioning
I Fu, Y., Klimkowski, K. J., Rodin, G. J., Berger, E., Browne,J. C., Singer, J. R., van der Greijn, R. A., Vemaganti, K. S.A fast solution method for the three-dimensional many-particle problems oflinear elasticity.Int. J. Num. Meth. in Eng., 42:1215–1229 (1998).
I Fujiwara, H.The fast multipole method for the integral equations of seismic scatteringproblems.Geophys. J. Int., 133:773–782 (1998).
I Fujiwara, H.The fast multipole method for solving integral equations of three-dimensionaltopography and basin problems.Geophys. J. Int., 140:198–210 (2000).
I Greenbaum, A.Iterative methods for solving linear systems.SIAM, Philadelphia, USA (1997).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 175 / 183
Preconditioning
I Greengard, L., Huang, J., Rokhlin, V., Wandzura, S.Accelerating fast multipole methods for the Helmholtz equation at lowfrequencies.IEEE Comp. Sci. Eng., 5:32–38 (1998).
I Greengard, L., Rokhlin, V.A fast algorithm for particle simulations.J. Comp. Phys., 73:325–348 (1987).
I Greengard, L., Rokhlin, V.A new version of the fast multipole method for the Laplace equation in threedimensions.Acta Numerica, 6:229–270 (1997).
I Gumerov, N. A., Duraiswami, R.Fast multipole methods for the Helmholtz equation in three dimensions.Elsevier (2005).
I Hackbusch, W., Nowak, Z. P.On the fast matrix multiplication in the boundary element method by panelclustering.Numerische Mathematik , 54:463–491 (1989).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 176 / 183
Preconditioning
I Hackbusch, W., Wittum, G. (eds.).Boundary elements: implementation and analysis of advanced algorithms.Wieweg, Braunschweig (1996).
I Jiang, L. J., Chew, W. C.A mixed-form fast multipole algorithm.IEEE Trans. Antennas Propagat., 53:4145–4156 (2005).
I Kellogg, O.D.Fundations of potential theory (1929).
I Kolk, K., Weber, W., Kuhn, G.Investigation of 3D crack propagation problems via fast BEM formulations.Comp. Mech., 37:32–40 (2005).
I Kupradze, V.D.Dynamical problems in elasticity.North-Holland , p. 259 (1963).
I Kurz, S., Rain, ; O., Rjasanow, S.The adaptive cross approximation technique for the 3D boundary elementmethod.IEEE Transactions on Magnetics, 38:421–424 (2002).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 177 / 183
Preconditioning
I Lage, C., Schwab, C.Wavelet Galerkin algorithms for boundary integral equations.SIAM J. Sci. Comput., 20:2195–2222 (1999).
I Liu, Y., Nishimura, N.The fast multipole boundary element method for potential problems: a tutorial.Engng. Anal. with Bound. Elem., 30:371–381 (2006).
I Liu, Y., Nishimura, N., Otani, Y., Takahashi, T., Chen, X. L.,Munakata, H.A fast boundary element method for the analysis of fiber-reinforced compositesbased on a rigid-inclusion model.ASME J. Appl. Mech., 72:115–128 (2005).
I Margonari, M., Bonnet, M.Fast multipole method applied to the coupling of elastostatic BEM with FEM.Computers and Structures, 83:700–717 (2005).
I Martinsson, P. G., Rokhlin, V.A fast direct solver for boundarry integral equations in two dimensions.J. Comp. Phys., 205:1–23 (2005).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 178 / 183
Preconditioning
I Nedelec, J. C.Acoustic and electromagnetic equations.Applied mathematical sciences (vol. 144). Springer Verlag (2001).
I Nemitz, N., Bonnet, M.Topological sensitivity and FMM-accelerated BEM applied to 3D acousticinverse scattering.Engng. Anal. with Bound. Elem. (2008).
I Nishimura, N.Fast multipole accelerated boundary integral equation methods.Appl. Mech. Rev., 55:299–324 (2002).
I Rokhlin, V.Rapid solution of integral equations of classical potential theory.J. Comp. Phys., 60:187–207 (1985).
I Rokhlin, V.Rapid solution of integral equations of scattering theory.J. Comp. Phys., 86:414–439 (1990).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 179 / 183
Preconditioning
I Rokhlin, V.Diagonal forms of translation operators for the Helmholtz equation in threedimensions.Appl. Comp. Harmonic Anal., 1:82–93 (1993).
I Rokhlin, V.Sparse diagonal forms for translation operators for the Helmholtz equation intwo dimensions.Appl. Comp. Harmonic Anal., 5:36–67 (1998).
I Saad, Y.Iterative methods for sparse linear systems.SIAM (2003).
I Saad, Y., Schultz, M.H.GMRES: a generalized minimal residual algorithm for solving nonsymmetriclinear systems.SIAM J. Sci. Stat. Comput., 7:856–869 (1986).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 180 / 183
Preconditioning
I Sakuma, T., Yasuda, Y..Fast multipole boundary element method for large-scale steady-state soundfield analysis. Part I: setup and validation.Acta Acustica united with Acustica, 88:513–525 (2002).
I Schmindlin, G., Lage, C., Schwab, C.Rapid solution of first kind boundary integral equations in R3.Engng. Anal. with Bound. Elem., 27:469–490 (2003).
I Song, J., Lu, C.C., Chew, W.C.Multilevel fast multipole algorithm for electromagnetic scattering by largecomplex objects.IEEE Trans. Antennas Propag., 42:1488–1493 (1997).
I Sylvand, G.La methode multipole rapide en electromagnetisme : performances,parallelisation, applications.Ph.D. thesis, ENPC (2002).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 181 / 183
Preconditioning
I Takahashi, T., Kawai, A., Ebisuzaki, T.Accelerating boundary-integral equation method using a special-purposecomputer.Int. J. Num. Meth. in Eng., 66:529–548 (2006).
I Takahashi, T., Nishimura, N., Kobayashi, S.A fast BIEM for three-dimensional elastodynamics in time domain.Engng. Anal. with Bound. Elem., 28:165–180 (2004).
I Tausch, J.Sparse BEM for potential theory and Stokes flow using variable order wavelets.Comp. Mech., 32:312–3185 (2003).
I Wang, P. B., Yao, Z. H..Fast multipole DBEM analysis of fatigue crack growth.Comp. Mech., 38:223–233 (2006).
I Yarvin, N., Rokhlin, V.Generalized Gaussian quadratures and singular value decompositions of integraloperators.SIAM J. Sci. Comput., 20:699–718 (1997).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 182 / 183
Preconditioning
I Ying, L., Biros, G., Zorin, D.A kernel-independent adaptive fast multipole in two and three dimensions.J. Comp. Phys., 196:591–626 (2004).
I Yoshida, K.I.Applications of fast multipole method to boundary integral equation method .Ph.D. thesis, University of Kyoto (2001).
I Yoshida, S., Nishimura, N., Kobayashi, S.Application of fast multipole Galerkin boundary integral equation method toelastostatic crack problems in 3D.Int. J. Num. Meth. in Eng., 50:525–547 (2001).
Marc Bonnet (POems, ENSTA) Methode d’elements de frontiere 183 / 183