{"id":2873,"date":"2025-01-12T13:25:36","date_gmt":"2025-01-12T13:25:36","guid":{"rendered":"https:\/\/nx365.ai\/2025\/01\/12\/modern-methods-of-integral-calculation\/"},"modified":"2025-05-27T09:38:18","modified_gmt":"2025-05-27T09:38:18","slug":"modern-methods-of-integral-calculation","status":"publish","type":"post","link":"https:\/\/nx365.ai\/en\/2025\/01\/12\/modern-methods-of-integral-calculation\/","title":{"rendered":"Modern Methods of Integral Calculation"},"content":{"rendered":"\n<h3 class=\"wp-block-heading has-text-align-center\"><strong>Modern Methods of Computing Integrals with Singularities in the Boundary Element Method <\/strong><\/h3>\n\n<p>A research team from the Netrix S.A. Research and Development Center and the WSEI Academy in Lublin, consisting of Dr. hab. Eng. Tomasz Rymarczyk and Prof. Eng. Jan Sikora, presented a groundbreaking approach to the numerical integration of functions with logarithmic singularities in the context of the Boundary Element Method (BEM). In the published paper <em>&#8220;Some More on Logarithmic Singularity Integration in Boundary Element Method,&#8221;<\/em> the authors introduce innovative computational techniques that significantly enhance the precision and efficiency of engineering simulations, particularly in acoustic applications.      <\/p>\n\n<p>The primary aim of the research was to analyze the effectiveness of two approaches to handling singularities in boundary integrals. The first method, the singularity-ignoring technique, involves the standard use of an increased number of Gauss integration points. Although simple to implement, this technique is associated with high computational costs, especially for meshes containing higher-order elements. An alternative is the singularity subtraction technique, which allows for the precise separation of the singular component from the regular part. This solution enables much greater computational accuracy while simultaneously reducing the number of numerical operations, making it particularly attractive for the analysis of complex boundary problems.     <\/p>\n\n<div style=\"height:30px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n<figure class=\"wp-block-image aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"658\" height=\"672\" src=\"https:\/\/nx365.ai\/wp-content\/uploads\/2025\/05\/1-14.jpg\" alt=\"\" class=\"wp-image-2147\" srcset=\"https:\/\/nx365.ai\/wp-content\/uploads\/2025\/05\/1-14.jpg 658w, https:\/\/nx365.ai\/wp-content\/uploads\/2025\/05\/1-14-294x300.jpg 294w\" sizes=\"(max-width: 658px) 100vw, 658px\" \/><figcaption class=\"wp-element-caption\"><em><em>Example of discretization of a homogeneous domain using second-order boundary elements with node and element numbering.<\/em><\/em><\/figcaption><\/figure>\n\n<div style=\"height:30px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n<figure class=\"wp-block-image aligncenter size-full\"><img decoding=\"async\" width=\"652\" height=\"599\" src=\"https:\/\/nx365.ai\/wp-content\/uploads\/2025\/05\/2-8.jpg\" alt=\"\" class=\"wp-image-2149\" srcset=\"https:\/\/nx365.ai\/wp-content\/uploads\/2025\/05\/2-8.jpg 652w, https:\/\/nx365.ai\/wp-content\/uploads\/2025\/05\/2-8-300x276.jpg 300w\" sizes=\"(max-width: 652px) 100vw, 652px\" \/><figcaption class=\"wp-element-caption\"><em>Area of interest and distribution of equipotential lines for different excitation frequencies. <\/em><\/figcaption><\/figure>\n\n<div style=\"height:30px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n<p>W pracy om\u00f3wiono r\u00f3wnie\u017c wp\u0142yw r\u00f3\u017cnych parametr\u00f3w fizycznych i geometrycznych na skuteczno\u015b\u0107 ca\u0142kowania. Dotyczy to m.in. cz\u0119stotliwo\u015bci fali akustycznej oraz obecno\u015bci ostrych kraw\u0119dzi w analizowanym obszarze. W kontek\u015bcie zagadnie\u0144 akustycznych opisywanych r\u00f3wnaniem Helmholtza, zastosowanie metody odejmowania osobliwo\u015bci umo\u017cliwi\u0142o uzyskanie \u015bredniego b\u0142\u0119du wzgl\u0119dnego (MRE) na poziomie 1\u20135%, nawet przy wysokich cz\u0119stotliwo\u015bciach. Dodatkowo, badania obj\u0119\u0142y zar\u00f3wno problemy Dirichleta, jak i szeroki zakres zagadnie\u0144 z zakresu propagacji fal.     <\/p>\n\n<p>These results have direct implications for engineering practice. The Boundary Element Method (BEM) is applied in many fields: from the design of acoustic systems, through electromagnetic field analysis, to the modeling of thermal and mechanical phenomena. The developed computational techniques not only increase the precision of simulations but also accelerate the design process, enabling faster and more reliable testing of different configurations.   <\/p>\n\n<p>A particularly interesting aspect is the ability to analytically calculate the singular component in the case of Laplace&#8217;s equations, which significantly speeds up integration procedures. Such solutions contribute to the further development of numerical tools, improving their quality and usefulness in modern engineering.  <\/p>\n\n<p>The full version of the article is available at:<br\/><a href=\"https:\/\/yadda.icm.edu.pl\/baztech\/element\/bwmeta1.element.baztech-e320d24a-a6a3-4e26-84a5-6f7ff67b0a11\/c\/Rymarczyk_Some_1_21.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">link to publication<\/a><\/p>\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Modern Methods of Computing Integrals with Singularities in the Boundary Element Method <\/p>\n","protected":false},"author":6,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[30,33],"tags":[],"class_list":["post-2873","post","type-post","status-publish","format-standard","hentry","category-news-ai-en","category-scientific-platform"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.1 - 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