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Seminar on Industrial Problems

ISDP: Inverse Shape Design Procedure

Seminar on Industrial Problems

ISDP: Inverse Shape Design Procedure

May 28, 1999

Presented
by:

** Gary
S. Strumolo **

Scientific
Research Laboratory

Ford Motor Company

10:10 am RealAudio (80 kbps) RealAudio (28.8 kbps)

Most computational fluid dynamics (CFD) analyses today follow the

The method is currently limited to potential
flow in two dimensions, although the procedure can be extended
to three. Sample problems will illustrate its capability, accuracy,
and ease of use.

Most computational fluid dynamics (CFD) analyses
today follow the *direct method*: specify the shape of
an object and CFD will compute the velocity or pressure along
its surface. This report will show how to perform the *inverse*
operation; namely, how to specify a velocity or pressure distribution
and compute the shape of a body needed to achieve this distribution.
The procedure, called ISDP, is fairly robust and can obtain
the desired shape even when the initial guess is far from the
final one. It works by migrating a set of vortex elements, whose
strengths are related to the desired distributions, until they
reach spatial positions consistent with them. Often the final
shape is reached with only a few dozen iterations, each of which
is computed directly without any matrix inversion. Two types
of boundaries are allowed: free and fixed. Free boundaries are
ones where the velocity/pressure is known but the shape isn't.
Fixed boundaries are ones where the shape is prescribed but
the velocity/pressure is unknown.

Broadly speaking, there are two types of inverse methods:

- Iterative use of a direct method
- True inverse methods

The first deals with successive guesses at the final shape. At each iteration, the latest shape is analyzed using the direct method and its actual surface velocity or pressure distribution is compared to the desired one. The error is used to construct a correction to the geometry for the next iterate. This approach is one step above trial-and-error, however, with the "corrections" hopefully providing some rationale for making shape changes. In contrast to this, the ISDP is a true inverse method, where the final shape is determined directly through a series of iterations.

We'll begin by describing the procedure for a purely free boundary case. Three shapes of increasing complexity will be sought: a circular cylinder, a spoiler-type wing, and the centerline profile of the Taurus. We will then describe how this technique can be modified to handle mixed boundary (free and fixed) conditions. This procedure has been awarded a US patent. We conclude with a discussion on stability considerations and the extension of the procedure to three dimensions.

Let's consider the case of free boundaries alone.
The problem can be stated as follows: determine the shape of
an object such that the flow over it results in a prescribed
distribution of pressure or velocity. Since the flow is assumed
to be potential-like, thus pressure (here represented by the
dimensionless pressure coefficient *C _{p}*)

Thus, *C _{p}* equals one at stagnation
points (V = 0) and zero when the local velocity equals that
of the free stream. Higher local velocities yield negative

The procedure employed is best illustrated using the following diagram:

**Figure 1.** *Steps
in inverse shape design for the free boundary case* [1].

The surface velocity *v _{s} *is
chosen as a function of distance along the proposed body, say,
from the leading to the trailing edge. We begin by making an
initial guess for the shape and dividing it into a series of

where is the external
free stream flow and D*q_{m}
*is the self-induced convection velocity
of element

The profile slope is measured clockwise from the horizontal, as shown in the following diagram:

**Figure 2.*** Illustration
of profile slopes [1].*

With these velocity components in hand, we can
realign the upper and lower surface of our object by treating
them as flexible chains with rigid, straight-line links. Each
link is aligned to be parallel to the velocity direction at
that point (step *d*). This realignment proceeds along
both the upper and lower surfaces from the leftmost point of
the profile, which is held fixed throughout the iterations.
Numerical round-offs in the calculations may cause the trailing
edge computed by marching along the upper surface to be different
from that obtained by the lower surface. Since the shape must
be closed for a converged solution to exist, a two-part correction
is applied at this point. First, the upper and lower surfaces
are rotated so that the lines connecting the fixed point to
the (possibly) two trailing edge points coincide. Then the surfaces
are stretched/shrunk so that the distance from leading to trailing
edge is maintained (steps *d,e*). This stretching/shrinking
of the profiles means that the initial guess need not be a small
perturbation from the final shape. Indeed, we will show that
a solution can be obtained even with the two far apart.

To assist in the rapid convergence of the algorithm,
a relaxation factor *f *is used. This quantity, between
zero and one, regulates how much of the newly-computed shape
is to be used after an iteration, i.e.,

If *f* is one, then the newly-computed shape
is used in its entirety as the new guess. If *f* is less
than one, the new shape is a combination of the previous shape
and the newly-computed iterate. At the end of an iteration the
velocity is then computed to be

For the iterations to stop this should agree with the desired velocity distribution to within a prescribed tolerance. Ideally, the final shape should have it's velocity/pressure distribution computed directly and compared to the desired one as a last check.

We will illustrate this procedure with a few examples. The first two have analytical solutions to provide the velocity/pressure distributions while the last is computed numerically. This saves us the need to back check our results via a direct solver.

While this may seem to be a simple problem, it is actually quite a challenge because of the shape near the stagnation points. It is well known that the velocity along the surface of the cylinder under potential flow is given by:

where *q
* is the polar angle measured from the
front stagnation point. The initial shape chosen was a thin
ellipse with a major-to-minor axis ratio of five. Both the upper
and lower surfaces were divided into 20 segments by simply marking
the ellipse every nine degrees. At the center of these segments
we assigned a velocity according to the above formula. The relaxation
factor *f *was set to 0.5. Figure 3 shows the results of
our computation; keep in mind that the desired shape is a unit
radius circle. The blue line shows the predicted shape after
30 iterations (square markers). The agreement with a circle
is extremely good. Given the initial guess, this example shows
that the initial and final shapes can be quite different; the
algorithm will still seek out the correct shape. It also illustrates
an *expansion solution*, where the initial guess expands
outward to reach the final shape.

In the previous calculation we used 20 segments to define each of the upper and lower surfaces. A reasonable question to ask at this point is how sensitive is the result to the "mesh" used. To answer this we re-ran the calculation using only eight segments per upper/lower surface (a reduction of 2.5 times). The number of iterations and relaxation factor was kept the same as before. The results are combined in figure 3.

**Figure 3***. 30^{th}
iterate using 20 divisions (solid line, squares) and 8 divisions
(dashed line, triangles).*

For both meshes, the markers still lie on the correct final shape! This clearly illustrates the robustness of the algorithm.

The next level of complexity is provided by wing
designs. Aerofoils of this type can be used for rear-deck spoilers.
We chose a special set of wings for which the shape and velocity/pressure
distributions are known analytically. To provide the mathematical
basis for our analysis, we will begin by illustrating the Joukowski
transformation, which converts a circle in the *z* plane
to an aerofoil in the *z*
plane:

**Figure 4.*** Joukowski's
transformation.*

The equation for the aerofoil in the *z*
plane is given by

where the polar coordinates (*r,**q
*) of the circle in the *z* plane
can be expressed as

The angles are defined by

If curve *C* is arranged to pass through
the singular point *B* at (*a*,0) then the aerofoil
will have a cusped trailing edge, which is not representative
of practical aerofoils. The Joukowski transformation can, however,
be used to produce aerofoils with a rounded trailing edge. To
achieve this we introduce an offset *e*3
and set

By picking values for *r*_{0}, *e*1,
*e*2,
and *e*3
(hereafter referred to as set A) we can create different wing
shapes. Putting this all together leads to the following expression
for the surface velocity on the aerofoil:

where the bound vortex strength G*
*is given by

The wing shape derived by picking set A = (0.25, 0.02, 0.03, 0.01) is:

**Figure 5.*** Sample
wing used in our analysis (inverted to resemble a rear end spoiler
orientation).*

We can use the same wing profile but change the free stream angle of attack slightly to give quite different surface velocity (or pressure) distributions over the same object shape.

Let's begin with the 5° case. As with the
circular cylinder, we start with an initial guess. However,
this time our guess will be larger than the final shape, illustrating
the algorithm's ability to also construct a *contraction solution.*
The ellipse chosen has a major axis equal to the final chord
length (which is assumed given) and a major-to-minor axis ratio
of two. Ten segments make up the upper and lower surfaces with
velocity values assigned as before.

Figure 6, shows the state of the computation after the 20th iteration. The desired curve is given by the blue solid line. The predicted shape is given by the red square markers. Except for an overshoot at the trailing edge (which is difficult to remove completely) the agreement with the desired distributions is very good:

**Figure 6.*** Comparison
of the computed shape (squares) to the desired one (line).*

As mentioned earlier, by changing the angle of
attack for the free-stream flow, we can obtain a different velocity/pressure
distribution *for the same object shape*. This provides
us with another test of the algorithm robustness: apply two
distinctly different velocity distributions and see if it can
generate the *same* profile shape. Using the analytical
pressure curves , we ran both cases and compared the two computed
profiles. These are shown in figure 7 along with the exact solution;
the match is quite good.

**Figure 7.*** Squares
- 5° case; triangles - 0° case; solid line - exact
solution.*

As a final test of the algorithm, we developed
a vehicle shape based on the centerline profile of the Taurus
and computed the *C _{p}* distribution along it's
upper and lower surfaces. Keep in mind that the distribution
used is for a two-dimensional body and should not be confused
with the centerline

For our vehicle test case, the upper and lower
surfaces were divided into 36 segments each and a circle was
chosen as the initial guess. After 20 iterations, and using
a relaxation factor of 0.9, we obtained the shape shown in figure
8. While there is some inaccuracy in capturing the *C _{p}
*distribution near the front stagnation point, the agreement
for both the upper and lower surfaces is quite good. The predicted
shape also shows remarkable agreement with the desired one,
particularly given the coarse mesh used for this type of body.

**Figure 8.*** Plot
showing Taurus shape recovered after 100 iterations using a
circle as the initial guess.*

In essence, the Taurus was "carved" out of the circular mold using the prescribed pressure distributions.

The previous algorithm assumes that all surface
shapes are to be computed, under the constraint of having a
certain velocity/pressure distribution. But consider the design
problem for a rear-end spoiler. We might want to know what the
shape of the spoiler should be when it lies next to a rear deck
whose shape and position is fixed. Thus, we would like to be
able to solve a mixed problem where the boundaries can be divided
into two classes. The first contains *free surfaces*, where
the velocity/pressure distribution is given but the shape is
to be determined. The second contains *fixed surfaces*,
where the shape is known but the velocity/pressure distribution
is unknown. Consider the following problem:

**Figure 9.*** Schematic
of rear-end spoiler problem showing fixed, free, and base surfaces.*

The fixed surface corresponds to that piece of the boundary that we know and are interested in. The base surface is added to the fixed surface so we have a closed surface over which potential flow can be computed.

How do we proceed with these mixed boundary conditions? The algorithm is as follows:

- Discretize the initial-guess free surface G
and the fixed surface G_{free}into_{fixed}*M*and*N*segments, respectively. Construct the base surface Gand append it to G_{base}so that we have two closed surfaces._{fixed} - Assign vortex strengths
permanently to G
._{free} - Solve the matrix problem on using a direct solver to obtain which yields potential flow over this composite surface.
- Compute the velocities on G
by using the previous procedure in the "all free surface" case but this time including the effects of the additional vortices ._{free} - Move the segments on G
according to the velocities obtained from (4). A few iterations at this point is recommended (i.e., looping over steps (4) and (5)) although it may not be necessary to iterate until convergence before proceeding to the next step._{free} - Loop back to (3) and repeat until convergence is achieved.

The net result of this procedure is the determination of the shape of the free surface with a prescribed velocity/pressure distribution along it as well as the computation of this distribution along the fixed portion of the boundary!

This new procedure enables us to consider more relevant problems such as the determination of a rear spoiler shape in the vicinity of a prescribed rear deck shape (2-D problem) and a side mirror shroud shape in the neighborhood of the A-Pillar, side glass (3-D problem).

The greatest danger in using this approach is prescribing a velocity/pressure distribution for which a solution does not exist, or one where the surfaces cross each other in an unphysical fashion. Assuming a reasonable solution exists, however, a general proof of uniqueness cannot be given, particularly when dealing with a numerical solution. But with additional constraints on the shape, it does appear that for bodies whose upper and lower surfaces are single valued and nonintersecting that a unique solution is obtained. The trick lies in how the slopes of the leading and trailing edges are defined.

For example, in the circular cylinder case we imposed that the two segments emanating from the leading edge did so at prescribed angles (nearly ± 90°). This is not an unreasonable or burdensome constraint since at a stagnation point the velocity is zero so the surface must be normal to the free stream flow just prior to it. A similar constraint could have been imposed at the trailing edge as well but we allowed it to remain free. Despite this, we can see from figures 3 and 4 that the correct profile was obtained. In both the spoiler-type wing and Taurus cases we imposed slope conditions at both the leading and trailing edges.

Despite the apparent robustness of the algorithm, it is possible for the iterations to sometimes stall or even diverge if the relaxation factor used isn't optimal. One improvement might be to use optimization theory to determine at each iteration a value for the factor that produces the greatest reduction in our residual error estimate. Thus, we could have an iteration sequence where the relaxation factor changes throughout, producing the most rapid convergence to the final shape.

This is the greatest challenge, but one which clearly must be met if we are to extend the scope of the present work. The direct solutions in three dimensions can be easily done using boundary integral techniques. The question is how do we define the vortex distribution for a three-dimensional body? Consider the following diagram for a section of a body using generalized coordinates:

**Figure 10.*** Surface
vorticity components of a vortex sheet *(*from
*[1]).

Whereas before we used one vortex along a segment of the two-dimensional body, we now need to use a pair of vortices for a surface patch. These two vortices are not independent, however, since we must have

Thus, only one is truly independent. We could
consider, say, *g*_{1}
as 'bound' vorticity giving rise to a 'shed' vorticity *g*_{2}.
These bound vortices might then be moved in a fashion similar
to that described earlier to reach the final desired shape.
This remains a subject for possible future research.

Reference:

[1] Lewis, R. I., **Vortex Element Methods for
Fluid Dynamic Analysis of Engineering Systems**, Cambridge
University Press, 1991.

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