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Finding the Optimal Place of Sensors for a 3-D Damped Wave Equation by using Measure Approach | ||
Control and Optimization in Applied Mathematics | ||
دوره 4، شماره 1، مهر 2019، صفحه 15-36 اصل مقاله (415.52 K) | ||
نوع مقاله: Research Article | ||
شناسه دیجیتال (DOI): 10.30473/coam.2020.48871.1126 | ||
نویسندگان | ||
Hajar Alimorad* 1؛ Alireza Fakharzadeh Jahromi2 | ||
1Jahrom University | ||
2Faculty of Mathematics, OR Department, Shiraz University of Technology, Shiraz, Iran | ||
چکیده | ||
In this paper, we model and solve the problem of optimal shaping and placing to put sensors for a 3-D wave equation with constant damping in a bounded open connected subset of 3-dimensional space. The place of sensor is modeled by a subdomain of this region of a given measure. By using an approach based on the embedding process, first, the system is formulated in variational form; then, by defining two positive Radon measures, the problem is represented in a space of measures. In this way, the shape design problem is turned into an infinite linear problem whose solution is guaranteed. In this step, the optimal solution (optimal control, optimal region, and optimal energy) is identified by a 2-phase optimization search technique applying two subsequent approximation steps. Moreover, some numerical simulations are given to compare this new method with other methods. | ||
کلیدواژهها | ||
3-D damped wave equation؛ Dissipation control؛ Radon measure؛ Search technique؛ Shape optimization | ||
عنوان مقاله [English] | ||
یافتن مکان بهینه حسگرها برای معادله موج میرای سه بعدی با استفاده از تقریب اندازهها | ||
نویسندگان [English] | ||
هاجر علیمراد1؛ علیرضا فخارزاده جهرمی2 | ||
1دانشگاه جهرم، دانشکده علوم پایه، گروه ریاضی. | ||
2دانشگاه صنعتی شیراز، دانشکده علوم پایه، گروه ریاضی. | ||
چکیده [English] | ||
در این مقاله، مساله تعیین شکل و محل بهینه نصب حسگرها برای معادلات موج سه بعدی با میرایی ثابت را در یک زیر مجموعه پیوسته از فضای سه بعدی، مدلسازی و حل میکنیم. محل قرار گرفتن حسگرها به وسیله زیردامنهای از یک ناحیه با اندازه مشخص مدلسازی شده است. با استفاده از یک تقریب بر مبنای روش نشاندن، ابتدا، دستگاه معادلات به صورت تغییراتی فرمولبندی شده؛ سپس، با تعریف دو اندازه رادن مثبت، مساله در فضای اندازهها بازنویسی خواهد شد. در این روش، مساله طراحی شکل به یک مساله برنامهریزی خطی نامتناهی تبدیل شده که وجود جواب آن تضمین شده است. در این مرحله، با استفاده از دو گام تقریب، جواب بهینه (کنترل بهینه، ناحیه و انرژی بهینه) با روش جستجوی دو مرحلهای تعیین میگردد. علاوه بر این، شبیه سازی عددی برای مقایسه روش جدید با دیگر روشها آورده شده است. | ||
کلیدواژهها [English] | ||
معادله موج سه بعدی, کنترل پراکندگی, اندازه رادن, روش جستجو, بهینهسازی شکل | ||
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