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ABSTRACT
In this article, the accuracy of the collocation Trefftz method (CTM) for solving two- and three-dimensional heat equations is investigated. The numerical solutions are approximated by superpositioning T-complete functions formulated using cylindrical harmonics. To avoid the ill-conditioning of the CTM, the characteristic lengths and the multiple-scale Trefftz method are adopted. The results reveal that for two-dimensional problems, the CTM can provide highly accurate numerical solutions, with the accuracy increasing with the order of the terms. For three-dimensional problems, highly accurate numerical solutions can be obtained using a certain order of terms, where the order is determined by performing an accuracy assessment.
Nomenclature
| aa | = | number of collocation points |
| A | = | aa × bb matrix [Eq. (12)] |
| b | = | aa × 1 vector [Eq. (12)] |
| b | = | aa × 1 vector [Eq. (20)] |
| bb | = | total number of T-complete functions |
| B | = | aa × bb matrix [Eq. (20)] |
| Iν | = | modified Bessel function of the first kind of νth order |
| I0 | = | modified Bessel function of the first kind of 0th order |
| Jν | = | Bessel function of the first kind of νth order |
| J0 | = | Bessel function of the first kind of 0th order |
| K | = | number of cavity |
| Kν | = | modified Bessel function of the second kind of νth order |
| K0 | = | modified Bessel function of the second kind of 0th order |
| M | = | order of the T-complete function |
| n | = | outward normal direction |
| = | outward normal vector | |
| N and O | = | order of the T-complete function |
| R | = | characteristic length |
| Rl | = | multiple scale characteristic length |
| u | = | unknown |
| x | = | bb × 1 vector [Eq. (20)] |
| y | = | bb × 1 vector [Eq. (12)] |
| Yν | = | Bessel function of the second kind of νth order |
| Y0 | = | Bessel function of the second kind of 0th order |
| ΓD | = | Dirichlet boundary |
| ΓN | = | Neumann boundary |
| ρ | = | radial distance |
| Ω | = | object domain |
| ∇2 | = | Laplacian |
Nomenclature
| aa | = | number of collocation points |
| A | = | aa × bb matrix [Eq. (12)] |
| b | = | aa × 1 vector [Eq. (12)] |
| b | = | aa × 1 vector [Eq. (20)] |
| bb | = | total number of T-complete functions |
| B | = | aa × bb matrix [Eq. (20)] |
| Iν | = | modified Bessel function of the first kind of νth order |
| I0 | = | modified Bessel function of the first kind of 0th order |
| Jν | = | Bessel function of the first kind of νth order |
| J0 | = | Bessel function of the first kind of 0th order |
| K | = | number of cavity |
| Kν | = | modified Bessel function of the second kind of νth order |
| K0 | = | modified Bessel function of the second kind of 0th order |
| M | = | order of the T-complete function |
| n | = | outward normal direction |
| = | outward normal vector | |
| N and O | = | order of the T-complete function |
| R | = | characteristic length |
| Rl | = | multiple scale characteristic length |
| u | = | unknown |
| x | = | bb × 1 vector [Eq. (20)] |
| y | = | bb × 1 vector [Eq. (12)] |
| Yν | = | Bessel function of the second kind of νth order |
| Y0 | = | Bessel function of the second kind of 0th order |
| ΓD | = | Dirichlet boundary |
| ΓN | = | Neumann boundary |
| ρ | = | radial distance |
| Ω | = | object domain |
| ∇2 | = | Laplacian |