Numerical Solution of the Optimization Issue with Restrictions for the Processes of Acoustic Wave’s Diffraction
L.V. Illarionova1, T.V. Kozhevnikova2, S.A. Pogorelov3
1
Candidate of Physical and Mathematical Sciences, numerical methods of mathematical physics (NMMP )laboratory, Computing Center ofFar-Eastern
Branch of the Russian Academy of Science (CC FEB RAS),
[email protected],2
Research scientist, information technologies laboratory, Computing Center ofFar-Eastern Branch of the Russian Academy of Science (CC FEB RAS),
3
Junior research assistant, numerical methods of mathematical physics (NMMP) laboratory, Computing Center ofFar-Eastern Branch of the Russian Academy of
Science (CC FEB RAS)
Abstract
The research is devoted to solving the optimal control problem for the equations of diffraction of acoustic waves by three-dimensional inclusion.It minimizes the deviation of the pressure field in the inclusion from a specific interval, due to sound sourceschanges in the external environment.
External boundary value problems for partial differential equations are the mathematical model of the process. In our research, we offer an algorithm for the numerical solution of the problem subject to limited set of controls. Namely, it imposes a restriction on the power of sources with which one can control the acoustic field.
The solution to the problem can be represented as a linear combination of the solutions of direct diffraction problems and unknown coefficients. We use the coordinate descent to find the required coefficients.
If we solve direct problems by potential theory, the diffraction problem reduces to a mixed system of weakly singular boundary integral Fredholm equations of the first and second kind on the inclusion surface. The approximation of integral equations by a system of linear algebraic equations is carried out by dividing the unit on the surface, associated with a system of nodal points, and also consistent with the discretization order of the method of averaging weakly singular kernels of integral operators. Multiple integrals arising during discretization are calculated analytically. This allows one to obtain explicit formulas for approximating boundary integral operators with singularities in kernels and use them to calculate the coefficients of systems of linear algebraic equations.
At the same time, preliminary triangulation of the surface is not required.
We ran numerical experiments and mathematical modeling of the diffraction process of acoustic waves. When solving a problem on a computer, parallel computations are used.
Keywords: acoustic waves, diffraction issue, numerical solution of control problems.
1. Introduction
The work is devoted to the solution of optimization diffraction issues of acoustic waves. Such issues are encountered in flaw detection,ocean acoustics and atmosphere, and geophysics. The mathematical model of the process is external boundary value issues for partial differential equations.
We used finite-difference and projection-grid methods of numerical solution to solve it [1–11], as well as numerical algorithms based on the transformation of the original differential issues to equivalent integral equations [12–17].
Current research is a continuation of articles [15,16,18,19] devoted to the development of the indirect method of integral equations for the numerical solution of direct and optimization diffraction issues.
The research considers a convex optimization diffraction issue of stationary acoustic waves in a homogeneous medium with an inclusion.
Objective assignment
Suppose that in a space𝑅3, filled with a uniform isotropic medium, there is a homogeneous bounded isotropic inclusion Ω𝑖 with a connected boundary𝑆. PutΩ𝑒 = 𝑅3∖ Ω𝑖 and denote it by𝜌𝑖(𝑒),𝑐𝑖(𝑒),𝛾𝑖(𝑒) the density, propagation velocity of acoustic vibrations, and absorption coefficient in Ω𝑖(𝑒).
Suppose that there are sound sources in the regionΩ𝑒. Sound waves propagate in space and, reaching inclusion, are scattered on it. As a result, reflected waves appear in the regionΩ𝑒, and transmitted waves appear inΩ𝑖.
Consider the following problem: by changing the sound sources in Ω𝑒 to minimize the deviation of the pressure field in Ω𝑖 (or on some subset𝑄 ⊂ Ω𝑖) from some required.
At the same time, the change in sound sources should not be ―large‖. Mathematically, itcanbeformulatedasfollows [16].
Find functions 𝑓: 𝑆 → 𝐶 (control) and F𝑖: Ω𝑖 → 𝐶, F𝑒: Ω𝑒 → 𝐶 (state) satisfying the following conditions
ΔF𝑖+ 𝑘𝑖2F𝑖 = 0 in Ω𝑖, (1)
ΔF𝑒+ 𝑘𝑒2F𝑒 = 0 in Ω𝑒, (2)
F𝑖− F𝑒 = 𝑔, 𝑝𝑖𝜕𝛷𝑖
𝜕𝒏 − 𝑝𝑒𝜕F𝑒
𝜕𝒏 = 𝑓 to 𝑆, (3)
𝜕F𝑒
𝜕|𝑥|− 𝑖𝑘𝑒F𝑒 = 𝑜 |𝑥|−1 as |𝑥| → ∞, (4) 𝐽(F𝑖, 𝑓) =1
2 F𝑖− F𝑑 2
𝑄
𝑑𝑥 +𝜆 2 |
𝑆
𝑓 − 𝑓𝑑|2 𝑑𝑠 → min, 𝑓
∈ 𝐾.
(5)
Here 𝒏 = 𝒏(𝒙) is the normal unit vector to the surface 𝑆 (directed towardsΩ𝑒), 𝑘𝑖(𝑒)2 =𝜔(𝜔 + 𝑖𝛾𝑖(𝑒))
𝑐𝑖(𝑒)2 , 𝑝𝑖(𝑒)= 1
𝜌𝑖(𝑒)𝜔(𝜔 + 𝑖𝛾𝑖(𝑒)),
𝜔 — wave circular frequency;𝑔, F𝑑, 𝑓𝑑 —complex-valued functions, 𝐾 — convexsetoffunctions, definedon𝑆 (admissible control set), 𝜆 — fixed nonnegative real number.
We propose an algorithm of the numerical solution of problems (1)-(5) on condition that the set𝐾. When solving a problem on a computer, parallel computations are used.
2. Materials and methods
Theoretical methods.The research contains methods of functional analysis (theory of generalized functions) and computational mathematics (theoretical foundations of numerical methods for solving systems of differential and algebraic equations, partial differential equations, and potential theory), theory of inverse problems.
Equipmentandsoftware.During the development and research of algorithms, methods of computational mathematics and methods of the theory of optimal control were used.
The software for the numerical solution of problems was created in Fortran-90 and is a console application designed to work on multiprocessor computing systems. The compiler is Intel Fortran Compiler. The software in question runs on the computing cluster of the Far-Eastern Branch of the Russian Academy of Science (http://hpc.febras.net/).
The computing cluster of the Far-Eastern Branch of the Russian Academy of Science is a heterogeneous system with a total capacity of 1520 Gigaflops (consists of 1 control and 17 computing nodes based on Intel Xeon and AMD Opteron processors). A total of 168 computing cores are available. A control network is the InfiniBand network,based on Gigabit Ethernet technology.
The computing subsystem of the cluster is based on three types of nodes. The first type includes 4 Sun x6440 blade servers, each of which is based on four six-core AMD Opteron 8431 processors, is equipped with an InfiniBand controller and has 96GB of RAM. The peak performance of each such node is 230 Gigaflops.
3. Results
3.1 Solution algorithm (1)–(5).
Let𝑒1, … , 𝑒𝑀be a system of functions linearly independent over the field𝑅 defined on the surface𝑆.
𝐾𝑀 = 𝐾 ∩ Span {e1, … , eM}.
WhereSpan {e1, … , eM}is alinear shell function𝑒1, … , 𝑒𝑀over the field𝑅.
Let𝑈𝑀be sets of three(F𝑖, F𝑒, 𝑓), comply with (1)–(4), but 𝑓 ∈ 𝐾𝑀.
Let us look onto the next problem. Find a function(F𝑖(𝑀), F𝑒(𝑀), 𝑓(𝑀)) ∈ 𝑈𝑀such as
∀(F𝑖, F𝑒, 𝑓) ∈ 𝑈𝑀 𝐽(F𝑖(𝑀), 𝑓(𝑀)) ≤ 𝐽(F𝑖, 𝑓). (6) The correctness of the problem and the convergence of the algorithm are proved in [2].
FunctionsF𝑖(𝑒)(𝑀)we will search in the form:
F𝑖(𝑒)(𝑀)= Ψ𝑖(𝑒)(0) + Ψ𝑖(𝑒), WhereΨ𝑖(𝑒)(0)isa solution to the diffraction issue,
ΔΨ𝑖(𝑒)(0) + 𝑘𝑖(𝑒)2 Ψ𝑖(𝑒)(0) = 0 in Ω𝑖(𝑒), Ψ𝑖(0)− Ψ𝑒(0)= 𝑔, 𝑝𝑖𝜕𝛹𝑖(0)
𝜕𝒏 − 𝑝𝑒𝜕𝛹𝑒(0)
𝜕𝒏 = 0 by 𝑆,
𝜕𝛹𝑒(0)
𝜕|𝑥| − 𝑖𝑘𝑒Ψ𝑒(0)= 𝑜 |𝑥|−1 by |𝑥| → ∞,
(7)
AndnewunknownΨ𝑖(𝑒)and𝑓(𝑀)must be complied with the formula
ΔΨ𝑖(𝑒)+ 𝑘𝑖(𝑒)2 Ψ𝑖(𝑒) = 0 in Ω𝑖(𝑒), (8)
Ψ𝑖− Ψ𝑒 = 0, 𝑝𝑖𝜕𝛹𝑖
𝜕𝒏 − 𝑝𝑒𝜕𝛹𝑒
𝜕𝒏 = 𝑓(𝑀) by 𝑆, (9)
𝜕𝛹𝑒
𝜕|𝑥|− 𝑖𝑘𝑒Ψ𝑒 = 𝑜 |𝑥|−1 by |𝑥| → ∞, (10)
1
2 Ψ𝑖− Ψ𝑑 2
𝑄
𝑑𝑥 +𝜆 2 |
𝑆
𝑓(𝑀)− 𝑓𝑑|2𝑑𝑠 → min, 𝑓(𝑀)∈ 𝐾𝑀. (11) HereΨ𝑑 = F𝑑 − Ψ𝑖(0).
LetfunctionsΨ𝑘,𝑖, Ψ𝑘,𝑒 (𝑘 = 1, 𝑀) be solutions of the next problem:
ΔΨ𝑖(𝑒)(0) + 𝑘𝑖(𝑒)2 Ψ𝑖(𝑒)(0) = 0 in Ω𝑖(𝑒), Ψ𝑖(0)− Ψ𝑒(0)= 0, 𝑝𝑖𝜕𝛹𝑘,𝑖
𝜕𝒏 − 𝑝𝑒𝜕𝛹𝑘,𝑒
𝜕𝒏 = 𝑒𝑘 by 𝑆,
𝜕𝛹𝑘
𝜕|𝑥|− 𝑖𝑘𝑒Ψ𝑘= 𝑜 |𝑥|−1 by |𝑥| → ∞.
(12)
Enter𝜑:𝑀→ 𝑅according to the formula 𝜑(𝜉) =1
2 𝜉𝑘
𝑀
𝑘=1
Ψ𝑘− Ψ𝑑
2
𝑄
𝑑𝑥 +𝜆
2 𝜉𝑘
𝑀
𝑘=1
𝑒𝑘− 𝑓𝑑
2
𝑆
𝑑𝑠 And set𝐾𝜉 = 𝜉 = (𝜉1, … , 𝜉𝑀) ∈𝑀: 𝑀𝑘=1𝜉𝑘𝑒𝑘 ∈ 𝐾𝑀 .
Then the solution of (8)–(11) is compiled by the formula:
𝑓(𝑀)= 𝜉𝑘
𝑀
𝑘=1
𝑒𝑘, Ψ𝑖 = 𝜉𝑘
𝑀
𝑘=1
Ψ𝑘,𝑖, Ψ𝑒 = 𝜉𝑘
𝑀
𝑘=1
Ψ𝑘,𝑒, Where(𝜉1, … 𝜉𝑀) = arg min𝜉∈𝐾𝜉𝜑 (𝜉).
Inthearticle [19] the algorithm described above was numerically implemented for the case when 𝐾 = 𝐿2(𝑆) (i.e., there are no control restrictions).
In our research, we consider the case when
𝐾 = 𝑓 ∈ 𝐿2(𝑆): |𝑓| ≤ by 𝑆 , (12)
Whereis agiven positive constant
Hypothesis|𝑓| ≤ imposes a limitation on the power of sources with which you can control the acoustic field.
Letfunctions{𝑒𝑘}comply with the formula |
𝑀
𝑘=1
𝑒𝑘| = 1, max
𝑥∈𝑆 | 𝑒𝑘(𝑥)| = 1 𝑘 = 1, 𝑀. (13)
It is not difficult to prove, that𝐾𝜉isn -cube:
𝐾𝜉 = 𝜉 ∈𝑛: |𝜉𝑘| ≤ 𝑘 = 1, 𝑀 .
Tofindunknowncoefficient𝜉𝑘we will use the coordinate descent method[20].
Configuration of coordinate descent method
1. Chooseaninitialpoint𝜉0= (𝜉10, 𝜉20, … , 𝜉𝑀0) ∈ 𝑅𝑀and𝜀 > 0 (determines the posterior error estimate).
2. Consider𝜉 = 𝜉0.
3. Godownthe1stvariable, i.e. exchange𝜉1to𝜂, where𝜂is a function minimum point𝜂 → 𝜑(𝜂, 𝜉2, … , 𝜉𝑀)under constraints|𝜂| ≤ .
4. Go down the 2nd, 3rd the same way…, 𝑀variable. So we get a new approximating=
(𝜉11, 𝜉21, … , 𝜉𝑀1)
5. Ifmax1≤𝑗 ≤𝑀 𝜉𝑗1− 𝜉𝑗0 ≤ 𝜀, then the algorithm is finished, approximate solution is𝜉1. If not, weconsider𝜉0= 𝜉1and move to the step 2.
3.2 Conditions of numerical experiments
1. Ω𝑖 = 𝑄is a ball of unit radius centered at the origin.
2. The set of admissible controls K is determined by the formula (13).
3. To solve the direct diffraction problems, we used the algorithm described in [16]. It consists of the following. By methods of potential theory, the diffraction problem reduces to a mixed system of weakly singular boundary integral Fredholm equations of the first and second kind on the inclusion surface. The approximation of integral equations by a system of linear algebraic equations is carried out by dividing the unit on the surface, associated with a system of nodal points, and also consistent with the discretization order
of the method of averaging weakly singular kernels of integral operators. Multiple integrals arising during discretization are calculated analytically. This allows one to obtain explicit formulas for approximating boundary integral operators with singularities in kernels and use them to calculate the coefficients of systems of linear algebraic equations. At the same time, preliminary triangulation of the surface is not required. We denote by N the number of sampling points used in solving diffraction problems.
4. Let𝑀𝜙, 𝑀𝜃 ∈ 𝑁,
𝑀 = 𝑀𝜙 ⋅ 𝑀𝜃, 𝜙𝑗 =2𝜋𝑗
𝑀𝜙 , 𝑗 = 0, 𝑀𝜙−1, 𝜃𝑚 =𝜋𝑚
𝑀𝜃 , 𝑚 = 0, 𝑀𝜃.
As𝑒𝑘functions were selected, half of which in spherical coordinates(𝜌, 𝜙, 𝜃), (𝜌 = |𝑥|, 𝜙
— longitude, 𝜃 — latitude) are determined by formulas (―hat‖ functions in coordinates(𝜙, 𝜃)):
𝑒𝑘(𝜙, 𝜃) = 𝑤 |𝜙𝑗 − 𝜙|2+ |𝜃𝑚 − 𝜃|2 By𝑘 = 𝑗𝑀𝜃+ 𝑚, 𝑗 = 0, 𝑀𝜙 − 1, 𝑚 = 0, 𝑀𝜃.
Here𝑤(𝑡) = exp 𝑡2/(𝑡2− 𝜋2) by |𝑡| < 𝜋
0 by |𝑡| ≥ 𝜋.
The second half of the 𝑒𝑘 functions is obtained from the ones defined above by multiplying by an imaginary unit.
3.3 Test calculation and numerical simulation Define𝛿∗=1
2 |𝑄 F𝑖− F𝑑|2𝑑𝑥 +𝜆
2 |𝑆 𝑓 − 𝑓𝑑|2𝑑𝑠,
𝛿 = 𝛿∗
1 ∥ 𝛷2 𝑑 ∥2+ 𝜆 2 ∥ 𝑓𝑑 ∥2⋅ 100, Example 1.Initial data
𝜌𝑖 = 5, 𝑐𝑖 = 1, 𝛾𝑖 = 0.02, 𝜌𝑒 = 3, 𝑐𝑒 = 0.5, 𝛾𝑒 = 0.05, = 5, F𝑑(𝑥) = exp( 𝑖𝑘𝑖𝑥3), 𝑓𝑑 = 0, 𝜆 = 0.
Figure 1 shows graphs of the dependence of 𝛿 on the number M (the number of functions𝑒𝑘) for various values of 𝑁 (the number of sampling points of the diffraction problem).
Fig.1. F𝑑 = exp( 𝑖𝑘𝑖𝑥3), а)𝜆 = 0, b) 𝜆 = 5
Example 2. In contrast to Example 1, F𝑑 𝑥 = 0. Figure 2 shows graphs of the dependence 𝛿∗of the number M for various values of N.
Fig.2. F𝑑 = 0а)𝜆 = 0, b) 𝜆 = 5
Figures 1 and 2 show that the error at different values of the parameter differs slightly. The number of basic functions has a greater effect on the error than the number of sampling points of direct diffraction problems.
Example3. The optimal control problem (1)-(5) is considered with the following initial data
F𝑑 = sin(𝜋𝑥1) , 𝑓𝑑 = 0, 𝑔 = 0,
𝜌𝑖 = 5, 𝑐𝑖 = 1, 𝛾𝑖 = 0.02, 𝜌𝑒 = 3, 𝑐𝑒 = 0.5, 𝛾𝑒 = 0.05.
The values of the parameters 𝜆and are indicated below.
Figure 3 shows the projective curves of the functions |F𝑑|и|F𝑖|for different values of the parameter𝜆andon the interval (а) |𝑥2| ≤ 0.45, 𝑥1,3 = 0.4; (б) |𝑥1| ≤ 0.45, 𝑥2= 0.4, 𝑥3= 0.
Fig.3. Projective function curves|F𝑑|и|F𝑖|for example 3.
Figure 4 showslevel lines and projective function interval |F𝑖|on the square|𝑥1,2| ≤ 0.45, 𝑥3= 0by (а) 𝜆 = 0, = 5 , (b) 𝜆 = 5, = 5, (c) 𝜆 = 0, = 1.
Fig.4 Level lines and projective surface |F𝒊|for example 3 by (а) 𝜆 = 0, = 5 , (b) 𝜆 = 5, = 5, (c) 𝜆 = 0, = 1.
Figures 3and 4 allow you to judge the influence of intervals𝜆andon the field of optimal pressure. Fields resulting from optimization are close to the functionF𝒅for small interval values𝜆and repeats the forms of𝜆. Decreasingoftheinterval𝜆reduces the difference |F𝑑− F𝑖|. Increasingoftheinterval , i.e. the weakening of restrictions on the power of sources leads to a decrease in the deviation of the calculated field from the specified.
Example4. (Acoustic attenuation task)The optimal control issue (1)-(5) is considered with the following initial data
F𝑑 = 0, 𝑓𝑑 =𝑒𝑥𝑝 𝑖𝑘𝑒 𝑥 − 𝑦
𝑥 − 𝑦 , 𝑦 = 2,0,0 , 𝑔 = 0, 𝜌𝑖 = 5, 𝑐𝑖 = 1, 𝛾𝑖 = 0.02, 𝜌𝑒 = 3, 𝑐𝑒 = 0.5, 𝛾𝑒 = 0.05.
From a physical point of view, this means that it is required to ―extinguish‖ the sound field in the inclusion created by an external point source.
Figure 5 hasprojective function curves |F𝑑|and|F𝑖|at different parameter values𝜆andon the interval (а) |𝑥2| ≤ 0.45, 𝑥1,3 = 0.4; (b) |𝑥1| ≤ 0.45, 𝑥2,3 = 0.
Fig.5. Projective function curves|F𝑑|and|F𝑖|for example 4.
Figure 6 haslevel lines and projective surface of the function|F𝑖|on the square|𝑥1,2| ≤ 0.45, 𝑥3= 0by (а) 𝜆 = 0.1, = 5 , (b) 𝜆 = 0.1, = 0.5, (c) 𝜆 = 1, = 5.
Fig. 6Level lines and projective surface|F𝒊|for example4by (а) 𝜆 = 0.1, = 5 , (b)
𝜆 = 0.1, = 0.5, (c) 𝜆 = 1, = 5.
Example5. (Acousticattenuationtask) Inspiteoftheexample 3 F𝒅= exp( 𝒊𝑘𝒊𝑥𝟑)(acoustic field is created by a plane wave).
Figure 7has projective function curves |F𝑑|and|F𝑖|at different parameter values𝜆on theinterval (а) |𝑥2| ≤ 0.45, 𝑥1,3 = 0.4; (b) |𝑥1| ≤ 0.45, 𝑥2,3 = 0.
Fig.7. Projective function curves|F𝑑|and|F𝑖|for example5.
Figure 8 haslevel lines and projective surface of the function |F𝑖|on the square|𝑥1,2| ≤ 0.45, 𝑥3 = 0при (а) 𝜆 = 0.1, = 5 , (b) 𝜆 = 0.1, = 0.5, (c) 𝜆 = 1,
= 5.
Fig.8Level lines and projective surface|F𝒊|for example5by (а) 𝜆 = 0.1, = 5 , (b) 𝜆 = 0.1, = 0.5, (c) 𝜆 = 1, = 5.
The source of sound waves is located to the right of the inclusion. The figures show that the greatest deviation from the required field occurs in that part of the inclusion, which is closer to the source. The smaller the value of λ and the greater the value of h, that is, the more you can change the field sources in the external environment, the closer the calculated field to zero.
4. Discussion
The results of test calculations and numerical experiments showed that the proposed algorithm converges and can be used to solve problems of diffraction of acoustic waves.
To carry out the calculations, the computing resources of the “Data Center of the Far-Eastern Branch of the Russian Academy of Science” [21] were used. This research was supported in through computational resources provided by the Shared Facility Center “Data Center of FEB RAS” (Khabarovsk) [22].
The bulk of the computation falls on the solution of direct diffraction problems. To speed up the algorithm, the parallelization method was applied. Since direct tasks are solved independently, it is effective to use the standard Fortran language function for the cycle of solving direct problems. The solution time depends on the number of cluster nuclei involved, while doubling the number of nuclei, the time decreases on average 1.5 times (until the saturation effect is achieved). Without the use of parallel calculations for N=616 and M=288, the error for test cases reached 1%.
The algorithm has linear computational complexity with respect to M and quadratic complexity with respect to N. The results of computational experiments showed that the proposed method accelerates the program up to 5 times. Moreover, for N=616 and M=2000, the error decreased to 0.1%. This means that the method has a second order of accuracy with respect to M.
Nevertheless, the usage of parallel computing can effectively solve the issue with greater accuracy.
5. Conclusion.
The main results of the work are as follows:
1. An algorithm for solving the optimization issue of diffraction of acoustic waves with restrictions was developed and tested.
2. The developed algorithm for solving the issue was tested on model problems of damping the sound inside the inclusion with limited power sources, with which you can control the acoustic fields.
3. Numerical experiments were conducted on a hybrid computing cluster based on the OpenPOWER architecture.
Acknowledgements
The research was supported in through computational resources provided by the ―Shared Facility Center\Data Center of FEB RAS‖ (Khabarovsk).
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