where is the number of influencing nodes (including the source node for which the n equation is written) and is the potential, i.e. The coefficient are p ps=p(xs, ys). αi’s evaluated from the boundary integral equation (3.31) and RBF interpolation, which is explained in Section 3.3. Note that the right hand side of (A.1) is zero since the equation is written for a node entirely within the domain, i.e. there is no acoustic sourcing term (unlike equation (3.34)). In the following, we will refer to the harmonic evolution theorem for dispersion error measurement in order to interpret the potential values, in (A.1) in terms of the potential at the source node, pi, (i=1,2,…,n), ps. According to harmonic evolution theorem, the potential at any point can be written i in terms of the potential at point s; (A.2) though the coefficients will be left un-evaluated for now (the explicit expressions τi are given in equation (3.48)). Using the relation (A.2), equation (A.1) can be as; (A.3) Equation (A.3) has the following interpretation. For an acoustic problem within the domain the rows of the system matrix for the nodes lying entirely within the Ω, domain are identical. Further, for a non-trivial solution of the problem, i.e. non-zero values of the following equation is being solved numerically; ps, (A.4) where the coefficients and are functions of the wavenumber. Therefore, one can αi τi solve an equation of type (A.4) by using root-finding algorithms, for one source node in order to find out the numerical wavenumber. Hence, the discrepancy between only, the numerical wavenumber and the exact wavenumber can be determined in order to
where n is the number of influencing nodes (including the source node for which the equation is written) and p is the potential, i.e. ps=p(xs, ys). The coefficient αi’s are evaluated from the boundary integral equation (3.31) and RBF interpolation, which is explained in Section 3.3. Note that the right hand side of (A.1) is zero since the equation is written for a node entirely within the domain, i.e. there is no acoustic sourcing term (unlike equation (3.34)). In the following, we will refer to the harmonic evolution theorem for dispersion error measurement in order to interpret the potential values, pi, (i=1,2,…,n), in (A.1) in terms of the potential at the source node, ps. According to harmonic evolution theorem, the potential at any point i can be written in terms of the potential at point s; (A.2) though the coefficients τi will be left un-evaluated for now (the explicit expressions are given in equation (3.48)). Using the relation (A.2), equation (A.1) can be as; (A.3) Equation (A.3) has the following interpretation. For an acoustic problem within the domain Ω, the rows of the system matrix for the nodes lying entirely within the domain are identical. Further, for a non-trivial solution of the problem, i.e. non-zero values of ps, the following equation is being solved numerically; (A.4) where the coefficients αi and τi are functions of the wavenumber. Therefore, one can solve an equation of type (A.4) by using root-finding algorithms, for one source node only, in order to find out the numerical wavenumber. Hence, the discrepancy between the numerical wavenumber and the exact wavenumber can be determined in order to
Thesis_Hakan_Dogan.docx