An Efficient Method for Time-Domain

7/30/2019 An Efficient Method for Time-Domain

1/40

An Efficient Method for Time-Domain

Low-Speed Aerodynamics

THEORY AND EX AM PL ES

David EllerDepartment of Aeronautics and Vehicle Engineering

Royal Institute of TechnologySE - 100 44 Stockholm, Sweden

December 2005

Abstract

A boundary element method for time-domain simulations of flexible

aircraft configurations has been developed. Due to the use of unstruc-

tured surface meshes and element clustering, modeling effort is reduced

and computational efficiency for complex geometries significantly im-

proved. A time-domain approach is chosen to enable the simulation ofunsteady problems involving nonlinear interaction phenomena.

Introduction

In recent years, computational methods in fluid dynamics have made remark-

able progress in terms of modeling accuracy, computational efficiency and ro-

bustness [1, 2]. While linear aerodynamic methods are still in use, especially

for unsteady problems [3], non-linear field methods which solve the Euler or

the Reynolds-averaged Navier-Stokes equations are applied more and more. Even

time-accurate unsteady computations with moving and deforming field meshes

can be performed, although at considerable computational cost [4, 5, 6]. Despitecontinually increasing processing power, time-accurate unsteady simulations still

require the use of scarce high-performance computing resources and both exten-

sive modeling experience and considerable efforts in mesh generation to obtain

solutions with good accuracy [5, 7].

Traditional linear methods, on the other hand, only require moderate com-

putational efforts and simpler mesh generation. For unsteady applications, the

Doublet-Lattice Method (DLM, [8]) is routinely used and has proved its value in

1


2/40

A 2 D. Eller

flutter calculations, in the determination of unsteady aerodynamic coefficientsand flight control system design. The formulation in the frequency domain

makes it especially well suited for many of these problems. However, the sim-

plified geometry representation used in the DLM comes at a price.

As an example, it is rather difficult to use the DLM to obtain loads (i.e.

surface pressures) for a structural shell model, since the Doublet-Lattice Method

provides only pressure differences between upper and lower side of a flat lifting

surface. Another example is that, due to the downwash discontinuity at control

surface hinge lines caused by modeling only the mid-surface of the real geome-

try, control surface effects appear to depend strongly on the local discretization

near the hinge line [9]. Finally, while the computational cost for DLM analysis

of small problems is indeed moderate, many implementations show a ratherpoor scaling behavior, namely an increase of the computational cost with at

least the square of the problem size expressed in the number of grid points or

surface elements. The factor usually dominating the computational cost is the

computation of influence coefficients, one of which must be computed for each

pair of panels (or points, depending on the formulation), leading to quadratic

algorithmic complexity. This is hardly relevant for standard applications em-

ploying just a few hundred panels, but becomes a significant impediment when

geometric complexity mandates discretizations with more than ten thousand

surface elements.

To enable time-domain analysis of geometrically complex problems with

moderate computational effort, a new boundary element method has thereforebeen developed. Similar to traditional panel codes as far as the flow model is

concerned, the current implementation differs in several aspects:

An unstructured triangular mesh surface discretization in order to enableautomatic mesh generation without user interaction.

Piecewise linear, as opposed to constant, singularity strength distributionsfor better resolution of large pressure gradients.

Application of surface element clustering in order to improve computa-tional efficiency for large problems.

Unsteady formulation allowing structural deformations and large-scaleflight mechanics motion.

Object-oriented, multithreaded implementation to simplify coupling withother simulations and accelerate solutions on parallel computers.

While a number of panel methods exist and some are available in source code

form, adapting them for coupled unsteady analysis would have required more


3/40

An Efficient Method for Time-Domain Low-Speed Aerodynamics A 3

effort than a proper new implementation which also allows the above improve-ments to be integrated.

Governing equations

The primary aim of the aerodynamic method is the computation of the unsteady

pressure distribution on the body surface. Integration or interpolation of the

pressure distribution yields global forces and moments as well as local loads on

the structure. The flow in the remaining field, off the aircraft body, is not of

immediate interest for the applications in mind.

In the following, the governing relations are derived in order to determinethe pressure distribution on the surface of a moving body. Boundary conditions

to be fulfilled at the surface are discussed next, and finally, the solution method

based on a standard boundary integral formulation is presented.

Remark on notation Vectors with three components (position, velocity, etc.)

are marked by an arrow superscript (x). Variables in bold print (x) are usedfor long vectors resulting from spatial discretization of a continuous problem.

Surface pressure

Consider an infinitesimal fluid element in unsteady incompressible flow, sub-jected to a pressure field p = p(t,x,y,z) and to shear stresses ij . The firstsubscript of is the normal of the plane in which the shear stress acts, while

the second index gives its direction. Hence, xz is the stress acting in the direc-

tion of the z axis in the yz -plane. The velocity vector v has the components

u,v,w in the x, y , z direction, respectively. For heavier-than-air applications in

aeronautics, volume forces such as gravity are neglected, so that the momentum

balance in Cartesian coordinates becomes

u

t+ (uv) =

p

x+

xx

x+

yx

y+

zx

z(1)

vt

+ (vv) = py

+ xyx

+ yyy

+ zyz

(2)

w

t+ (wv) =

p

z+

xz

x+

yz

y+

zz

z. (3)

Furthermore, conservation of mass for the same fluid element leads to

t+ v = 0. (4)


4/40

A 4 D. Eller

For sufficiently large Reynolds numbers and streamlined bodies such as aircraftat low to moderate angles of attack, the influence of the viscous stresses ij is

limited to thin boundary layers at the body surface. For the external flow, shear

stresses are small and can be neglected. Dropping the shear stress terms and

inserting (4), equations (1) - (3) can be written in vector form according to

v

t+ v v =

p

, (5)

which is known as the Euler equation for the motion of an inviscid fluid.

Assuming that the velocity field is free of rotation, that is if

v = 0 (6)

holds everywhere, the Euler equation can be simplified using

v v =1

2

v2 v ( v) (7)

to take the formv

t+

1

2

v2

= p

. (8)

For attached flow about aircraft configurations, vorticity is assumed to exist only

in the thin vortex sheet emanating from the trailing edge of lifting surfaces. The

condition of irrotational flow does therefore not hold when applied through the

wake surface.

In irrotational flow, the velocity can be expressed as the gradient of a po-

tential field (t,x,y,z). Since the current method deals with moving aircraftbodies, it is convenient to write the velocity as a superposition of body motion

and wind according to

v(t,x,y,z) = (t,x,y,z) = (t,x,y,z) + vw(x, y , z), (9)

where vw is the steady wind velocity and is the potential field caused by themotion of the body in the fluid. This formulation is useful since it does not

require that the potential of the wind velocity field is known. Note that the

wind is assumed to vary in space, but not in time (steady wind), which simpli-fies the computation of surface pressures. This assumption may at first appear

rather restrictive when considering e.g. aircraft loads caused by turbulence. For

atmospheric conditions such as those which must be considered for certifica-

tion of civil aircraft [10, 11], the unsteadiness is caused by the aircraft moving

through a localized steady wind gradient transverse to the flight path at high

speed. Therefore, the capability to include arbitrary wind gradients is assumed

more important than the unsteady character of the wind field itself.


5/40


Substituting (9) into (8), the relation

t+

1

2 ( + vw)

2 +p

= 0 (10)

between potential and pressure is found, which, for an incompressible fluid with

= 0 can also be written as

t

E

+1

2( + vw)

2 +p

= 0 (11)

where the time derivative of the potential is taken in an inertial frame E. Re-

lation (11) is a vector equation stating that the gradient of the expression in

parentheses vanishes everywhere in the domain. This expression must hencebe constant in space. For applications to external flow problems, the domain

stretches to infinity, where the flow must be undisturbed by the presence of the

body. The boundary conditions at infinity can hence be expressed according to

(t) = 0, = 0, p = p, vw = vw,. (12)

Inserting values at infinity into (11) yields that

t

E

+1

2( + vw)

2 +p

=

1

2v2w, +

p

(13)

is valid throughout the domain. This form contains the time derivative of the

potential in the Eulerian frame, that is the rate of change at a fixed point in

space. The current method is primarily concerned with the potential at the

surface of the body, which moves with a local velocity vb with respect to the

Eulerian reference frame. Therefore, the inertial time derivative of the potential

is replaced with its derivative in the moving body frame B according to

t

E

=

t

B

vb , (14)

where vb is the velocity of the body surface. In this form, it is sufficient to

compute the time derivative of the potential at a point on the moving body and

the gradient of the potential at the body surface, evaluated in the inertial frame

E. Using the above, (13) can be formulated as

p p

=

t

B

+ vb 1

2( + vw)

2 +1

2v2w,. (15)

As soon as the potential at the body surface is known, the unsteady surface

pressure can be computed from (15). The body surface velocity and wind field

must be known to solve for potential and pressure.


6/40

A 6 D. Eller

Boundary conditions

Due to the absence of viscous stress, only a solid wall boundary condition of

the form

n v = 0, (16)

where n is the surface normal, can be enforced with the simplified flow model.

This condition states that there is no flow through the body surface. In order

to model propulsion systems or to approximate the effect of boundary layers,

a prescribed transpiration velocity vt can be postulated additionally. Further-

more, the motion of the body must be accounted for, as illustrated in Figure 1.

Considering the local velocity vb of a point on the surface and the prescribed

Figure 1: Sketch to illustrate boundary conditions.

transpiration, the boundary condition at the wall can be expressed as

n ( + vw) = n (vb + vt) . (17)

Here, all velocities and the gradient of the potential are computed in the non-

moving frame E. The normal direction n, body velocity vb and transpiration

velocityvt are, in general, time-dependent. Introducing the normal derivative

n= n , (18)

the above can be written according to

n= n (vb + vt vw) . (19)

This is the transpiration boundary condition which must be fulfilled on the

whole surface and at all times.

At the trailing edge of lifting surfaces, an additional condition is enforced.

If it were not imposed, the flow would curve around the sharp trailing edge


7/40


and reach very high velocities due to the sharp angle. As such a flow patterncontradicts experimental evidence, an empirical condition is introduced which

states that the fluid must not flow around the trailing edge (Kutta, [12]). The ex-

perimental observation alone does not directly motivate a specific mathematical

definition, so that several different formulations exist. Here, a purely kinematic

condition is chosen, which is illustrated in Figure 2. Consider two points at

Figure 2: Notations for the boundary condition at the trailing edge.

the trailing edge designated by the subscripts u (upper surface) and l (on the

lower wing surface). The outward-pointing surface normal vectors are denoted

nu and nl, and the corresponding local surface gradients u and l. Using the

normalized vectorte which points along the trailing edge itself, a trailing edgenormal vector ne can be defined by setting

ne =1

2

nu te + te nl

. (20)

The vector ne is perpendicular to the trailing edge and bisects the angle between

the surface normals. The kinematic form of the Kutta condition is then imposed

by enforcing that the projections of the upper and lower trailing edge velocities

are equal, according to

ne (u + vw) = ne (l + vw) , (21)

which is equivalent to

ne (u l) = 0. (22)

Lacking a better formulation, this form of the Kutta condition is also used for

unsteady problems. Experimental evidence indicates that, for fast motion of

the trailing edge transverse to the airspeed vector, the kinematic condition is

no longer fulfilled [13, 14]. The experimenters also found that the validity of


8/40

A 8 D. Eller

the smooth outflow condition depends not only on the frequency, but also onthe magnitude of the transverse velocity, which suggests that non-linear effects

become important at high frequency.

Boundary integral equations

The governing equation for the potential itself is derived from the conservation

of mass (4). For an incompressible fluid, the density is constant in time and

space so that (4) simplifies to v = 0. Inserting (9) yields

() = + vw = 0. (23)

Assuming that the prescribed wind field fulfills incompressible continuity, i.e. vw = 0, the Laplace equation for the potential is obtained according to

2

x2+

2

y2+

2

z2= 0. (24)

To summarize, a solution of (24) which fulfills the boundary conditions

(19),(22) must be computed in order to obtain surface pressures by means of

(15). The potential only needs to be computed at the surface of the body, sothat a boundary element method (BEM) can be employed. In contrast to field

discretization methods, such as the finite-element or finite-volume method, the

BEM requires only a discrete representation of the boundary of the computa-

tional domain, which in this case is the body surface.

Boundary integral equations to solve the Laplace equation subjected to the

boundary condition (19) can be derived by means of Greens identity, as shown

e.g. by Katz and Plotkin [14, Chapter 3.2]. The solution in the domain of interest

is represented by a surface distribution of singularities which are fundamental

solutions to the Laplace equation and fulfill the farfield boundary condition

(12). Let the surface of the body be covered by a continuous distribution of

source () and doublet () singularities. The singularities are defined as

=

n

in

(25)

= i, (26)

where i is the potential on the inside surface of the body and the value ofthe potential on the outer surface. For any point on the surface, there exist hence

two singularity strengths and , but only one boundary condition, namely

(19). It is possible to specify only one of the above singularities, but using both

has additional advantages. If both and are employed, the transpiration

boundary condition can be fulfilled with many combinations of singularities.


9/40


Each of these yields a different distribution for the internal potential i. Tomake the problem unique, the single solution which fulfills i = 0 everywhereon the inner surface is chosen. With this particular choice, the singularities

simplify to

= (27)

and, as shown by Lamb [15, 40],

=

n. (28)

The above only holds for closed surfaces. Since both singularities are only

defined on the surface itself, the gradient of the potential on the surface is

obtained as

= t n, (29)

where t is the tangential gradient of the doublet density and n the unitsurface normal.

The advantage of enforcing the internal potential to vanish is that the source

strength distribution can be computed explicitly from (19) once the motion

and transpiration on the surface are known, and that the surface potential and

its gradient are readily computed from (27) and (29). With this formulation,

the actual computational problem is to determine the doublet strengths which

fulfill i = 0. This is a Dirichlet problem since the value of the velocity

potential is prescribed along the boundary of the domain. One of the benefitsof the formulation in terms of the inner potential is that the requirements

on the necessary continuity of the surface singularity densities can be relaxed.

According to Gray [16], a direct formulation in terms of the boundary condition

(19) would require that the tangential derivative of the singularity strength

is continuous for the surface integrals to be defined. In other words, and

would need to be defined as piecewise bi-cubic functions on the surface (Hermite

elements) which, in turn, makes the evaluation of the corresponding surface

integrals rather involved.

From the potential of source and doublet singularities, the influence on a

point on the inside of the body surface is obtained as

i = 1

4

SB

||r||

n

1

||r||

dSB, (30)

where r is the distance vector from the point P to the integration point on the

surface, and SB is the body surface [14]. The effect of the wake is considered in

the following.

At the trailing edge, the upper and lower side of the lifting surface meet at

a sharp angle. The kinematic Kutta condition (22) cannot be fulfilled unless


10/40

A 10 D. Eller

the velocity potential on the upper and lower surface can take different values.Hence, the upper and lower surface point must be distinct, although both share

the same geometric position. Because of the latter, the integral equations for

the internal velocity potential would be redundant if imposed for both points.

Instead, i = 0 is only used for one of the two points, while the potential equa-tion for the second is replaced by the Kutta condition in the form (22). Using

(29), the kinematic Kutta condition can be written in terms of the singularity

densities according to

ne (t,l t,u) = ne (unu lnl) , (31)

which is a linear equation in for each trailing edge vertex.A wake surface carrying doublet singularities is modelled starting at the

trailing edge and continues downstream. It can hence be understood as a con-

tinuation of the upper and lower surface potentials toward downstream infinity.

Physically, the wake surface can be regarded as a model of the thin vortex sheet

emanating from the trailing edge. The doublet densities of the wake surface

correspond to the difference of the surface potential between upper and lower

trailing edge point [14].

Including the effect of the doublet distribution on the wake surface SW,

the velocity potential at a point on the inner surface is defined by the integral

expression

4i =

SB

r+

n r

r3

dSB +

SW

n r

r3dSW, (32)

where the normal derivatives have been expanded. The length of the distance

vector r has been replaced byr for a more compact notation.

According to Kelvins law for incompressible inviscid flow [17], the material

time derivative of the circulation must vanish. The wake is not a material

surface and can hence not support any loads, it moves with the local fluid

velocity, which is illustrated in Figure 3. Any point in the wake, associated

with a certain wake strength, can thus be regarded as attached to the same fluid

particles at all times. Since vorticity (circulation) is related to the wake doubletdensity, Kelvins law requires that the doublet strength of a given point in the

wake must be constant in time. In a time-dependent problem, the wake doublet

density for a wake point thus remains the same once it has left the trailing

edge. Wake strengths w for downstream elements are therefore known from

previous times, so that only the currently shed doublets, i.e. the wake strength

at the trailing edge, is unknown. Since the latter is written as the difference

between upper (u) and lower (l) trailing edge surface doublet strengths, no


11/40


Figure 3: Front view of the wake surface behind a glider.

new unknown variables are introduced by the wake model. Finally, the equation

for vanishing internal potential can be written according to

SB

n rr3

dSB +St

(u l)n r

r3dSt

=

SB

rdSB

SW

wn r

r3dSW, (33)

where St is the part of the wake which carries the trailing edge doublet strengths,

and SW is the remaining wake surface, which was shed previously.

Numerical solution

In order to solve (32) for an arbitrary surface geometry, the integral equationis discretized using a surface mesh. Surface integrals are then evaluated as the

sum of integrations over all discrete elements. In the following, the spatial

discretization is described first. In the next section, an efficient approach for

the evaluation of the farfield integration is presented. Finally, some details of

the parallel implementation are mentioned.

Discretization

In order to facilitate the mesh generation for complex configurations, an un-

structured mesh with triangular elements is used. Source and doublet strengths

are distributed linearly over the surface elements. With linear instead of the morecommon constant singularity strengths, a better representation of the pressure

distribution is obtained, particularly in regions where the pressure gradient is

large. For the computation of surface pressures by means of (15), the tangential

derivative of the doublet density, t, is required. This term is constant overlinear triangular elements, while it is not even defined for a single element carry-

ing a constant . In most traditional panel methods, the gradient term is there-

fore computed by interpolation between adjacent surface elements [14, 18, 19].


12/40

A 12 D. Eller

More advanced techniques use higher-order polynomial approximation for thepotential, relying on a structured quadrilateral surface mesh [20].

Additionally, the use of linear doublet distributions in the wake allows an

accurate integration of the wake influence on the trailing edge points. An advan-

tage for unsteady problems is that the discrete wake surface can be constructed

as defined by the actual motion of the trailing edge without the need to cre-

ate a shifted or reduced first row of wake panels. The latter procedure must

be adopted if constant strength wake doublets are used in order to empirically

compensate for the otherwise excessive integration error [14, 21].

For a linear triangular element parameterized in and , the singularity

strengths over the element are

(, ) = 1 + (2 1) + (3 1) (34)

(, ) = 1 + (2 1) + (3 1), (35)

where the indices 1-3 designate vertex values of the singularities. The geometry

of the elements is also linear, i.e. the triangles are plane and their normals con-

stant. Vertex normals which are needed to evaluate the transpiration boundary

condition (19) are computed from triangle normals using the angular weighting

procedure of Thurmer and Wuthrich [22].

The boundary integral equation (32) can be solved by means of collocation

or Galerkin methods. Using collocation, (32) is enforced to vanish at a set

of collocation points on the surface. A Galerkin method, on the other hand,would minimize the surface integral of some norm of the internal potential.

In the present method, a collocation approach is implemented because it does

not require the additional level of surface integration needed for a Galerkin

approach.

Using linear singularity distributions on triangular elements results in two

unknown (source and doublet) variables per mesh node. The source strength

is computed using (28), so that one equation per vertex must be obtained by

means of (32). Hence, the mesh nodes themselves are selected as collocation

points.

The element integrations are performed in cylindrical coordinates with the

origin in the projection of the collocation point onto the triangle plane. Thedistance vector r which appears in the singular integrals can now be expressed

in cylindrical coordinates as

r = [rp sin , rp cos , h] , (36)

where rp is the distance between the integration point and the projection Q

of the collocation point P in the element plane, the in-plane angle and h


13/40


the distance of the collocation point above the element plane. The notation isillustrated in Figure 4. Singularity strengths depend linearly on the triangle pa-

rameters and , and can be expressed in terms ofrp and by transformation.

The resulting relations are linear in rp, but involve rational functions of sin .

The singularity strengths can thus be written schematically as

(rp, ) = rpm1() + m2() and (rp, ) = rps1() + s2(), (37)

where m and s are functions of only. Using these cylindrical transformations,

Figure 4: Influence coefficient integration by sector.

the doublet and source potential integrals for a fixed collocation point can beexpressed according to

= + (38)

4 =

r2ps1 + rps2

(r2p + h2)

1

2

drp d (39)

4 =

hr2pm1 + hrpm2

(r2p + h2)

3

2

drp d. (40)

For nonzero, but arbitrarily small h, i.e. a collocation point which is not exactly

on the surface, the above expressions are singular, but integrable with respect torp. Evaluation of the integral from rp = 0 to some upper limit R() yields

4 =

s1

2

RL +

h2

2ln

h

R + L

2+ s2 (L |h|)

d (41)

with

limh0

4 =

s1

2R2 + s2R d, (42)


14/40

A 14 D. Eller

and

4 =

2R

L ln

h

R + L

2

m1h

2+

h

|h|

h

L

m2

d, (43)

where the substitution

L =

R2 + h2 (44)

is used. The remaining integral with respect to is rather intricate, but regular

since the singularity is located at rp = 0, h = 0.

The remaining one-dimensional source potential integration of (41) can be

computed analytically. For the doublet potential (43), this is however much moredifficult. Therefore, the integration with respect to is performed numerically

using the standard 8-point Gauss integration rule [23]. Comparing accuracy for

different orders of the integration rule used for the -direction showed that four

points were not always sufficient. Gauss-integration with more than 5 points,

however, did no longer show any dependence on the order. Finally, the 8-point

integration rule was chosen to accommodate even badly shaped triangles.

As illustrated in Figure 4, the integration of the influence of a single sur-

face element on one collocation point is performed by summing three sector

integrals. Each sector is the triangular region between the projection Q of the

collocation point P and one of the triangle edges. For the case shown in Fig-

ure 4, the influence of the element would be obtained schematically by

4i =

Sa

() dSa +

Sb

() dSb +

Sc

() dSc. (45)

As already mentioned above, the numerical evaluation of the doublet poten-

tial integral is only valid for collocation points which are not located exactly

in the triangle plane. The mathematical reason for this is that the value of the

doublet influence integration is not defined uniquely on the surface itself. For

a discrete surface with edges, it evaluates to

limh04 = Ko for h > 0

Ki for h < 0 (46)

where Ko and Ki are the external and internal solid angles spanned by the

triangles at the collocation vertex. Because of this ambiguity, the integral can

only be computed for collocation points with |h| > 0. In an initial imple-mentation, the special case h = 0 was detected during the evaluation of (32),and the corresponding integral values were replaced by the known limit val-

ues from (46). When using this approach, it is not obvious how to treat the


15/40


single collocation point at the trailing edge, where the solid angle is not well-defined. Although a possible solution was found by defining this solid angle in

terms of the surface-wake continuation, doubts remained about the results for

strongly curved trailing edges. As the latter occurs often when control surfaces

are modeled using the current mesh deformation approach, another solution

was investigated.

A rather straightforward alternative is to displace the collocation point a

small distance into the body surface. An investigation of the behavior of the nu-

merical kernel integration for extremely small h was performed, for which some

results are presented in Figure 6. A primitive mesh of a four-sided pyramid,

shown in Figure 5, is created and the source and doublet potential influence

integrated for a collocation point which incrementally approaches the surface.The logarithmically scaled diagram in Figure 6 shows the development of the

Figure 5: Simple mesh for validation.

108

106

104

102

100

0.4

0.2

0

0.2

0.4

0.6

Normal distance

Potentialinfluence

upper doublet potentiallower doublet potentialupper source potentiallower source potential

Figure 6: Potential vs. distance.

integral terms for small normal distances. For this particular case, the critical

influence potential for a unit doublet converges towards a fixed value for dis-

tances h < 106le, where le is the edge length of the pyramid base. If h ischosen smaller than that values as small as 1018le were used the six mostsignificant digits of the computed potential do not change any longer. The

relative error with respect to the values from (46) is 2.7 106

for the insidepotential and 1.6 104 for the outside potential.

In the current implementation, the collocation point is displaced by 106

times the average length of the mesh edges incident on a vertex inside the surface.

Collocation points at the trailing edge are moved inside the body by the same

distance, using the trailing edge normal vector ne as defined in (20) to replace

the surface normal in this case.

Discretization of (32) by collocation yields a system of algebraic equations


16/40

A 16 D. Eller

which must be solved for the unknown surface doublet strengths . In contrastto most field methods, the resulting system features a dense matrix of influ-

ence coefficients. A method which computes the influence coefficients directly

requires O(n2v) operations, only to compute the coefficient matrix, and evenO(n3v) operations to factorize it if a dense direct solver is used, where nv isthe number of mesh vertices. Such a solution method would not scale well to

reasonably refined meshes of complex configurations, since the algorithmic cost

increases too quickly with mesh size.

Clustering

The algorithmic complexity of boundary element methods can be vastly im-proved by means of panel clustering methods [24, 25]. Clustering is based on

the observation that the value of the integrand in (32) decreases strongly with

increasing distance between collocation and integration point. It is therefore

possible to combine sets of panels into clusters and compute the respective po-

tential for equally clustered sets of sufficiently distant collocation points using

an efficient approximation.

Surface elements and collocation points are grouped into clusters by means

of a binary tree hierarchy as proposed by Lage [25]. The top-level (root) cluster

contains all elements, its child cluster two disjoint sets which are found by

subdividing the root set geometrically. Bebendorf [26] suggest to divide clusters

along a plane through the barycenter, which is oriented approximately normalto the first principal direction of the point cloud in the parent cluster. This

criterion is inexpensive to evaluate and creates a reasonably well balanced tree.

The subdivision is repeated recursively for all sibling trees until the leaf nodes

reach a certain minimum size, which can be a single element or collocation

point.

Using the binary cluster trees for surface elements (panel tree) and colloca-

tion points (vertex tree), the potential influence can be computed by descending

the trees. In each level, a separation criterion

= 2max(r1, r2)

||rc|| (r1 + r2)(47)

is evaluated. Here, r1, r2 are the radii of the panel and vertex cluster, and

rc is the distance between their centers. If is positive and smaller than a

user-defined limit, the influence of all the elements in the panel cluster on all

collocation points in the vertex cluster is computed using an efficient degenerate

kernel described below. For negative , or if the separation limit is exceeded,

the influence computation is passed to all combinations of the siblings of the

current panel and vertex tree nodes, unless the clusters are already leaf nodes, in


17/40


which case the influence is computed directly by looping over the elements andpanels in the respective cluster.

The cluster to cluster potential evaluation is performed by means of a Taylor

expansion of the kernel functions. As in most panel methods, the farfield

influence of a panel on a collocation point is first approximated as the potential

of point doublets and sources, which is far less involved to compute and makes

little difference in the farfield [14]. For a more readable notation, the kernel

functions (32) are abbreviated as k(x, y), where x replaces the location of thepoint singularity and y the collocation point. Then, the point singularity kernel

is expanded according to

k(x, y) = k(x, yc) +ml=1

lk

l!y l

x,yc

(y yc)l + Rm, (48)

where xc and yc are the centers of the panel and collocation point cluster, and

Rm is the remainder for expansion order m. As y is in fact a vector, the

derivatives of k should be interpreted as gradient, Hessian etc. of the kernel

with respect to the collocation point location. Given a panel cluster with npvertices and a collocation point cluster with nc points, the potential is evaluated

approximately according to

kl =

npi=0

lk

l!y l

xi,yc

(49)

k(xi, yj) ml=0

kl (yj yc)l . (50)

The algorithmic cost of computing the np nc influence coefficients in thecluster pair thus becomes proportional to m3(np + nc), while the cost of thedirect evaluation would be proportional to np nc. Further improvements can

be gained by means of the related multipole expansion [21, 25], for whichthe complexity of the cluster-to-cluster evaluation decreases to m2(np + nc).The multipole technique is hence particularly advantageous for high accuracy

expansions.

For the current implementation, expansion orders m of two and three are

used. An upper bound for the expansion error Rm for the source and doublet

kernel is found by comparing the evaluation of the expansion with the actual

value for the least favorable case. For orders one to three and source () and


18/40

A 18 D. Eller

doublet () singularities, the remainder terms are found to be less than

R(m = 1)

An Efficient Method for Time-Domain

Documents

Transcript of An Efficient Method for Time-Domain