finite element method +helmholtz

8/6/2019 Finite Element Method +Helmholtz

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wherefis given, u is an unknown function ofx, and u'' is the second derivative ofu with respect to x.

The two-dimensional sample problem is the Dirichlet problem

where is a connected open region in the (x,y) plane whose boundary is "nice" (e.g., a smooth manifold orapolygon), and uxx and uyy denote the second derivatives with respect to x and y, respectively.

The problem P1 can be solved "directly" by computing antiderivatives. However, this method of solving theboundary value problem works only when there is only one spatial dimension and does not generalize to higher-

dimensional problems or to problems like u + u'' = f. For this reason, we will develop the finite element method

for P1 and outline its generalization to P2.

Our explanation will proceed in two steps, which mirror two essential steps one must take to solve a boundary

value problem (BVP) using the FEM.

y In the first step, one rephrases the original BVP in its weak form. Little to no computation is usually required forthis step. The transformation is done by hand on paper.

y The second step is the discretization, where the weak form is discretized in a finite dimensional space.After this second step, we have concrete formulae for a large but finite dimensional linear problem whosesolution will approximately solve the original BVP. This finite dimensional problem is then implemented on a

computer.

Weak formulation

The first step is to convert P1 and P2 into their equivalents weak formulation. Ifu solves P1, then for any

smooth function v that satisfies the displacement boundary conditions, i.e. v = 0 at x = 0 and x = 1,we have

(1)

Conversely, ifu with u(0) = u(1) = 0 satisfies (1) for every smooth function v(x) then one may show that this u

will solve P1. The proof is easier for twice continuously differentiable u (mean value theorem), but may beproved in a distributional sense as well.

By using integration by parts on the right-hand-side of (1), we obtain


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(2)

where we have used the assumption that v(0) = v(1) = 0.

A proof outline of existence and uniqueness of the solution

We can loosely think of to be the absolutely continuous functions of (0,1) that are 0 at x = 0 and x =

1 (see Sobolev spaces). Such function are (weakly) "once differentiable" and it turns out that the symmetric

bilinear map then defines an inner product which turns into a Hilbert space (a detailed proof is

nontrivial.) On the other hand, the left-hand-side is also an inner product, this time on the LpspaceL

2(0,1). An application of the Riesz representation theorem for Hilbert spaces shows that there is a unique

u solving (2) and therefore P1. This solution is a-priori only a member of , but using elliptic

regularity, will be smooth iffis.

The weak form of P2

If we integrate by parts using a form ofGreen's identities, we see that ifu solves P2, then for any v,

where denotes the gradient and denotes the dot product in the two-dimensional plane. Once more can be

turned into an inner product on a suitable space of "once differentiable" functions of that are zero on

.We have also assumed that (see Sobolev spaces). Existence and uniqueness of the solution

Helmholtz equation

The Helmholtz equation, named forHermann von Helmholtz, is the elliptic partial differential equation

where 2 is the Laplacian, k is the wavenumber, and A is the amplitude.

Motivation and uses

The Helmholtz equation often arises in the study of physical problems involvingpartial differential equations

(PDEs) in both space and time. The Helmholtz equation, which represents the time-independent form of the


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original equation, results from applying the technique ofseparation of variables to reduce the complexity of theanalysis.

For example, consider the wave equation

Separation of variables begins by assuming that the wave function u(r, t) is in fact separable:

Substituting this form into the wave equation, and then simplifying, we obtain the following equation:

Notice the expression on the left-hand side depends only onr, whereas the right-hand expression depends onlyon t. As a result, this equation is valid in the general case if and only if both sides of the equation are equal to a

constant value. From this observation, we obtain two equations, one forA(r), the other forT(t):

And

where we have chosen, without loss of generality, the expression k

2

for the value of the constant. (It is equallyvalid to use any constant kas the separation constant; k2 is chosen only for convenience in the resulting

solutions.)

Rearranging the first equation, we obtain the Helmholtz equation:

Likewise, after making the substitution

the second equation becomes

where k is the wave vectorand is the angular frequency.


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Harmonic solutions

It is relatively easy to show that solutions to the Helmholtz equation will take the form:

which corresponds to the time-harmonic solution

for arbitrary (complex-valued) constants Cand D, which will depend on the initial conditions and boundaryconditions, and subject to the dispersion relation

We now have Helmholtz's equation for the spatial variable r and a second-orderordinary differential equationin time. The solution in time will be a linear combination ofsine and cosine functions, with angular frequency

of , while the form of the solution in space will depend on theboundary conditions. Alternatively, integraltransforms, such as the Laplace orFourier transform, are often used to transform a hyperbolic PDE into a formof the Helmholtz equation.

Because of its relationship to the wave equation, the Helmholtz equation arises in problems in such areas of

physics as the study ofelectromagnetic radiation, seismology, and acoustics.

The real part of the resulting field A, A is the solution to the inhomogeneous Helmholtz equation

Helmholtz Differential Equation

An elliptic partial differential equation given by

where is a scalar function and is the scalarLaplacian, or

where is a vector function and is the vector Laplacian (Moon and Spencer 1988, pp. 136-143).

When , the Helmholtz differential equation reduces to Laplace's equation.When (i.e., for imaginary), the equation becomes the space part of the diffusion equation.

The Helmholtz differential equation can be solved by separation of variables in only 11 coordinate systems, 10of which (with the exception ofconfocal paraboloidal coordinates) are particular cases of the confocal

ellipsoidal system: Cartesian, confocal ellipsoidal, confocal paraboloidal, conical, cylindrical, ellipticcylindrical, oblate spheroidal,paraboloidal,parabolic cylindrical,prolate spheroidal, and spherical coordinates

(Eisenhart 1934ab). Laplace's equation (the Helmholtz differential equation with ) is separable in the twoadditionalbispherical coordinates and toroidal coordinates.


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If Helmholtz's equation is separable in a three-dimensional coordinate system, then Morse and Feshbach (1953,pp. 509-510) show that

where . The Laplacian is therefore of the form

which simplifies to

Such a coordinate system obeys the Robertson condition, which means that the Stckel determinant is of the

form

finite element method +helmholtz

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