A methodology for determining the scattered Electromagnetic (EM) fields present for interconnected regions with common metasurface boundaries is presented. The method uses a Boundary Element Method (BEM) formulation of the frequency domain version of Maxwell's equations -which expresses the fields present in a region due to surface currents on the boundaries. Metasurface boundaries are represented in terms of surface susceptibilities which when integrated with the Generalized Sheet Transition Conditions (GSTCs), gave rise to an equivalent configuration in terms of electric and magnetic currents. Such a representation is then naturally incorporated into the BEM methodology. Two examples are presented for EM scattering of a Gaussian beam to illustrate the proposed method. In the first example, metasurface is excited with a diverging Gaussian beam, and the scattered fields are validated using a semi-analytical method. Second example concerned with a non-uniform metasurface modeling a diffraction grating, whose results were confirmed with conventional Finite Difference Frequency Domain (FDFD) method.

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The field of computational engineering and experimentation relies very heavily on methods of advanced and accurate model simulation of partial differential equations as found in heat flow, the wave equation, and electromagnetics. A majority of these methods use techniques such as Finite Difference or Finite Elements that require the meshing of the geometric region and knowledge of the connectivity and relationships between each segment. A newly proposed method, the Finite Cloud Method (FCM), removes the need for the onerous and sometimes difficult task of computing this mesh, instead uses shaping functions and a discretized set of partial differential equations based only on the placement of nodes [1]. The ability of the FCM to allow for the distribution of solution points to areas of complexity in a completely free manner could enable faster more accurate simulations. However, initial work has focused on materially homogenous problems and the extension of the technique to models composed of different materials with varying physical properties is needed for practical problems. This study presents a method of formulating the FCM equations such that they allow for the specification of varying material properties and is applied to the heat transfer equation. Results from the work have shown an ability to accurately model complex structures in 3-dimensions for both transient and steadystate solutions.Keywords: meshfree methods / finite cloud method / partial differential equations / Schlüsselwörter: netzfreie Methode / Finite Wolke Methode / Differentialgleichungen /

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