Q2. What is the way to determine continuity of heat flux?
Assumption of a standard polynomial basis on one side of an interface, along with explicit computation of the necessary changes to that basis that occur across the interface (through continuity matrix inversion), determines the piecewise polynomial functions that must be present in an expansion of temperature data near the interface.
Q3. How many nodes are supported by the RBF-FD stencils?
RBF-FDstencils that cross an interface or are close to the domain boundary contain 30 nodes and are supported by all polynomials up through 4th degree.
Q4. What is the way to enforce continuity of temperature along a curved interface?
To enforce continuity of temperature along a curved interface, after application of a given power k of the discrete form of the differential operator in (1), a local expansion for the interface shape itself (again, in terms of 'x ) can be inserted into entries for 'y .
Q5. What is the way to determine the weights of the RBFs?
Although a value of 0.4 in the numerator of (31) has proven reliable in solving the test cases presented here, additional accuracy could possibly be gained by decreasing the numerator to a smaller constant (as long as the resulting RBF-FD weights produce an acceptably stable solution).
Q6. What is the definition of support monomials in 2-D?
In 1-D, support monomials were characterized across an interface by enforcing discrete forms of (6) and (7) for constant expansion coefficients (and constant term time derivatives) of temperature and heat flux.
Q7. What is the way to determine continuity of temperature and heat flux?
k k t k tu u D uy y n (27)Enforcing linear relationships between the expansion coefficients of as described in the last two paragraphs ensures that continuity of temperature and heat flux are upheld to ( )pO h accuracyat ' 0y for a mildly-curved interface that is locally very well-represented by a linearapproximation.
Q8. What is the purpose of this paper?
Although the paper focuses on modification of the supplemental support polynomials to enforce interface continuity conditions, the authors also describe and implement a method for modifying the RBFs to help achieve the same goal.
Q9. What is the radius from the center point of the insulating ring?
The radius from the center point (0.5,0.5) and heat diffusivity are defined as follows:0.52 2( 0.5) ( 0.5)r x y (37)1/1500 (1/ 3000)sin(2 )sin(2 ) 0.349 0.35 1 x y if r otherwise (38)There is also a third, circular Dirichlet boundary at 0.05 to simulate a cooling unit inside the insulating ring.
Q10. What is the simplest way to test a 2-D interface?
The next 2-D test problem involves two mildly curved interfaces within a unit square that isperiodic in the -direction and closed at = 0 and = 1 (Figure 8).