Numerical simulation of high frequency acoustic and electromagnetic scattering

1. The University of Reading Department of Mathematics Numerical simulation of high frequency acoustic and electromagnetic scattering Stephen Langdon and Simon Chandler-Wilde k=10 k=20 Background Acoustic and electromagnetic wave scattering problems arise in numerous physical and industrial applications, such as the use of radar or sonar to determine properties of a distant object, or the use of ground penetrating radar to determine what can be found under the surface of the earth. Although the mathematical simulation of these problems has a long history, they continue to present a range of challenges. Many such problems can be formulated as the Helmholtz equation ∆u+k²u=0 in Rd \ Ω d=2,3 (1) supplemented with appropriate boundary conditions. Here k>0 (the wavenumber) is an arbitrary positive constant, proportional to the frequency of the incident wave. As k grows, corresponding to the incident field oscillating more rapidly, so the complexity of the solution of (1) increases. This is demonstrated in figure 1, where we show what happens when a time harmonic acoustic plane wave strikes a sound-soft obstacle, in this case a square, and is scattered. k=5 incident field scattered field correction to leading order behaviour Improving performance at high frequencies The computational cost of standard schemes will typically grow in direct proportion to k, leading to prohibitively large computing times when k is large. The reason for this is clear from figure 2. In order to accurately model a wave, a fixed number of elements (typically 5-10) is needed per wavelength. If the wavelength is short compared to the size of the obstacle (corresponding to high frequency / large k) then many such elements are required. The numerical simulation of many realistic problems is thus unattainable with current technologies. In order to achieve an accurate solution to (1) for large k with a reasonable computational cost, the oscillatory nature of the solution must be taken into account when designing computational methods. Recent research has focused on enriching the approximation space with highly oscillating functions, such as plane waves or Hankel functions, in order to accurately represent the scattered field. total field Fig.1 Scattering by a square • Galerkin method with special basis functions • Reformulate (1) as an integral equation (I + K)ψ=f, for the unknown ψ, where f and the integral operator K are given. • Subtract the leading order behaviour. Using ideas from the Geometric Theory of Diffraction we write the scattered wave as a combination of reflected and diffracted rays. The leading order behaviour of ψ then corresponds to the known reflected rays. Subtracting these leaves a similar integral equation for a new function Φ with a different right hand side g. • Determine regularity results for Φ. For example, for a problem of scattering by a convex polygon, we show that we can write Φ(s)= eiksv+(s) +e-iksv-(s) on each side of the polygon, with s denoting the distance from a corner, where k-n|v±(n)(s)|≤C(ks)-1/2-n for ks≥1, and k-n|v±(n)(s)|≤C(ks)-α-n for ks≤1, where the positive constant α<1/2 depends on the angle of each corner. This decay is shown in figure 3 – the corners occur for s/2π = 0,1,2,3,4, and Φ is highly peaked at these points. The rate of decay away from the corners increases with increasing k. • Galerkin boundary element method. Our scheme is to find Φn in the space Vn such that (Φn,Χ)+(KΦn,Χ)=(g,Χ) for all Χ in Vn. The key then is to choose an appropriate approximation space Vn, and to do this we use the regularity result above to design a graded mesh with smaller elements near the corners, so as to take advantage of the peaked behaviour of v±. The approximation space then consists of plane waves e±iks supported by piecewise polynomials on this graded mesh. Fig. 2 Modelling a wave Example 1: Scattering by a flat surface of varying impedance Combining plane waves with a non-uniform mesh, we recently presented a new numerical scheme for this problem for which the computational cost required to achieve a fixed level of accuracy remains fixed for all values of k>1. For this problem the total acoustic pressure one wavelength above the plane is shown in figure 4, with an obstacle of length 5(i), 10(ii), 20(iii), 40(iv), 80(v) and 160(vi) wavelengths. Even though the solution becomes more highly peaked as the ratio of obstacle length to wavelength grows, the computational cost required to accurately solve (1) does not change. Fig.3 Peaked behaviour of Φ near corners Example 2: Scattering by convex polygons Using a Galerkin boundary element method as described above, we have derived for this problem a scheme for which the computational cost required to achieve a fixed level of accuracy grows only logarithmically with respect to k. This compares well with the linear growth rate associated with standard schemes. For a problem of scattering of an acoustic plane wave by a sound soft triangle, we show in figure 5 the "computed correction", the difference between the total field and the leading order asymptotic behaviour. k=5 k=10 k=20 Fig.5 “computed correction” to leading order asymptotic behaviour, scattering by a triangle References S.N. Chandler-Wilde, S.Langdon and L. Ritter, A high-wavenumber boundary-element method for an acoustic scattering problem, Phil. Trans. R. Soc. Lond. A.362 (2003) 647-671 S. Langdon and S.N. Chandler-Wilde, A wavenumber independent boundary-element method for an acoustic scattering problem, SIAM J. Numer. Anal (to appear) Fig. 4 Acoustic pressure above an impedance plane, increasing frequency Research supported by the Leverhulme Trust