Analysis of geometric uncertainties in 3D thermo-fluid problems solved by RBF-FD meshless method

The tolerances of production processes can lead to uncertainties in the behaviour and in the features of the manufactured products. From the point of view of the design of engineering components it is therefore of valuable practical interest to be able to quantify such uncertainties as well as the expected, i.e., averaged, performances. Such uncertainty quantification is carried out in this work by means of the Non-Intrusive Polynomial Chaos (PC) method in order to estimate the propagation of geometrical uncertainties of the boundaries, i.e., when the boundaries are described by stochastic variables. Existing deterministic solvers can be used with the PC method because of its non-intrusive formulation, allowing an accurate and practical prediction of the random response through a simple set of deterministic response simulations. The Radial Basis Function Finite Differences (RBF-FD) method is employed as a black box solver for the computation of the required set of responses defined over deterministic boundaries. The RBF-FD method belongs to the class of meshless methods which do not require a computational mesh/grid, therefore its main capability is to easily deal with practical problems defined over complex-shaped domains. The geometrical flexibility of the RBF-FD method is even more advantageous when coupled to the Non-Intrusive PC method for uncertainty quantification since different deterministic solutions over different geometries are required. The applicability of the proposed approach to practical problems is presented through the prediction of geometric uncertainty effects for a steady-state forced convection problem in a 3D complex-shaped domain.


Introduction
Manufacturing tolerances are pervasive in almost every area of engineering and their impact can often be crucial: indeed, tolerances in production processes lead to an uncertain behavior in the performances of the manufactured products.These uncertainties should be quantified to some extent early in the design phase, especially when dealing with robust design.With this in mind, it is essential to develop efficient and accurate numerical methods aimed at quantifying such uncertainties [1].In this work we focus on the uncertainty quantification in the presence of geometric uncertainties [1][2] on the boundaries in 3D incompressible, laminar and steady-state thermo-fluid problems.
In this work the stochastic problem of uncertainty quantification is solved by means of the Non-Intrusive Polynomial Chaos method [3][4][5][6].The non-intrusive formulation [7] allows the use of existing, deterministic solvers as black boxes, i.e., without the need for any adjustment or modification: the assessment of the sought stochastic response is reduced to the computation of a limited set of deterministic simulations.The Polynomial Chaos formulation, on the other hand, minimizes the number of required deterministic responses, allowing a fast and accurate estimate of the sought stochastic response [4][5].The required simulations on deterministic geometries are computed in this work through the Radial Basis Function-generated Finite Differences (RBF-FD) meshless method [8].This choice is due to the geometric flexibility and the ability of RBF-FD meshless method to easily deal with complex-shaped domains since a simple distribution of nodes is required only.This numerical approach, along with many other meshless/meshfree techniques [9], is recently gaining more and more popularity in practical applications since the traditional mesh/grid required by wellknown mesh-based methods, e.g., Finite Volume or Finite Element methods, is no longer needed.Furthermore, the convergence rate of RBF-FD methods can be easily increased in order to provide high-fidelity numerical results with optimal computational effort [10].The coupling of Non-Intrusive Polynomial Chaos and RBF-FD method has already been successfully employed for the prediction of geometric uncertainty in 2D thermo-fluid problems [11], highlighting the practical feasibility of such approach.In this work the aforementioned approach is employed for the quantification of the statistical moments, i.e., expected value  and variance  2 , of the local and global Nusselt number for the forced convection inside a 3D pipe with wavy surfaces whose amplitude is described by two stochastic variables with Gaussian distribution.

Governing equations
Let us consider an incompressible fluid with density , kinematic viscosity , thermal diffusivity  and thermal conductivity .The resulting conservation equations of mass, momentum and energy are: In the above equations, length, velocity  = (, , ), time , pressure  and temperature T are made nondimensional by taking L,  0 , L/ 0 ,  0 2 and ∆ as reference quantities, respectively.Re =  0 / = 500 is the flow Reynolds number, Pr = / = 0.71 is the Prandtl number.
The maximum velocity at the inlet (see Appendix A) is chosen as the reference velocity  0 and the difference between the maximum temperature at the inlet and the temperature at the wavy surfaces is chosen as the reference temperature scale ∆.Because of the double symmetry of both the domain and the previous boundary conditions with respect to the  −  and the  −  planes, only one fourth of the previous domain is considered for the following numerical computations, i.e., ,  > 0. The symmetry conditions are the following: Graphical representations of the computational domain are shown in figure 2.

Tensorial-expanded Polynomial Chaos
Given a vector of  independent random variables  1 ,…, M ), under certain conditions [4] a random process  with finite second-order moments can be expanded as follows: where {φ  } is a suitable basis and   are the corresponding expansion coefficients.In this work random variables with gaussian distribution   ~N(0,1) will be considered only, for which the best choice for the basis {φ  } in terms of convergence rate for the sought statistical moments is given by Hermite polynomials (Hermite-Chaos).Indeed, 1D Hermite polynomials are orthogonal with respect to the gaussian probability density function (PDF), therefore the -dimensional tensorial-expanded Hermite polynomials are orthogonal with respect to the -dimensional gaussian PDF: The expansion (6), expressed in the Hermite-Chaos form and truncated to a finite number Q + 1 of terms, is: where   is the Hermite polynomial of degree .The statistical moments of , i.e., expected value/mean μ and variance 2 , can be computed from the previous approximation: In order to obtain the best approximation for μ and  2 , we note that equations ( 9) and ( 10) are weighted integrals with a gaussian weight function ().Therefore, the best choice for the  + 1 = = ( 1 + 1) ⋯ (  + 1) sample points, i.e., the points where  has to be evaluated, are the roots of the Hermite polynomial of degree   + 1 along each of the  = 1, ⋯ ,  dimensions, i.e., Gauss-Hermite quadrature.By using a collocation technique, the Hermite-Chaos expansion (8) is therefore made valid at each of these  + 1 sample points in order to obtain the  + 1 unknown coefficients   1 ⋯  .In other words, the Hermite-Chaos expansion can be seen as an interpolant matching the random process  at the sample points.

RBF-FD discretization and solution procedure
The RBF-FD meshless method [7] is employed to discretize the governing equations on meshless distributions of N nodes scattered over the domain and on the boundary.Local interpolants with polynomial degrees  = 2, 3, 4, requiring  = 20, 40, 70 stencil nodes, respectively, have been used.Implicit ∇ 4 hyperviscosity is employed to suppress advective and pressure instabilities [8,11].More details can be found in [12].
At each time step, the computation of velocity, pressure and temperature through equations ( 1)-( 3) is decoupled using a projection scheme with a three-level Gear scheme for the time discretization.Both the advective and diffusive terms are treated implicitly in equations ( 2)-(3), allowing the use of larger time steps to reach the steady-state solutions.BiCGSTAB iterative solver with an incomplete LU factorization as preconditioner are employed for the solution of the whole set of equations using a relative tolerance of 10 -9 .The steady-state convergence is declared when |Nu ̅̅̅̅ =1,2 /| < 5 • 10 −6 , where Nu ̅̅̅̅  are the average Nusselt numbers over the internal and external wavy surfaces.
The whole procedure is developed in Julia [13], a high-level, high-performance, dynamic programming language particularly suited for scientific computing.

Verification of the RBF-FD deterministic solver
The verification of the solver presented in subsection 3.2 is carried out by considering the analytical solution presented in Appendix A. The convergence curves are shown in figure 1 for node distributions with N ranging from 50,000 to 800,000 nodes.The root mean square (RMS) error over the whole domain for velocity and temperature is shown in figure 1a: the convergence rates of the RMS error for RBF-FD polynomial degrees  = 2, 3, 4 are 2.8, 3.2, 5.5, respectively, showing a great advantage in the use of higher polynomial degrees, both in terms of convergence rate and absolute magnitude of the error.On the other hand, the increase from P = 2 to P = 3 highlights only a moderate improvement in the accuracy.The flattening of the RMS error of T around 10 -8 for N > 300,000 nodes is due to the single precision accuracy in the generation of boundary nodes.The relative error for the average Nusselt numbers is shown in figure 1b, where the convergence rate and the absolute value of the error both improve by increasing the RBF-FD polynomial degree P, for which the convergence rate is approximately equal to P.

Deterministic simulations
In order to highlight the convergence properties of the deterministic RBF-FD solver in the case of the actual problem described in section 2, a convergence study is performed for node distributions with N ranging from 50,000 to 800,000 nodes and polynomial degrees  = 2, 3, 4.An example of node distribution with N = 200,000 nodes is depicted in figure 2a and 2b, while the computed temperature field is shown in figure 2c and 2d.From these figures it can also be noted how the node density is increased near the internal wavy surface highlighted in figure 2d, i.e., where the thermal boundary layer is thinner and more nodes are required to capture the gradients with accuracy.
The results of the convergence study are summarized in table 1 in terms of average Nusselt numbers Nu ̅̅̅̅ 1 and Nu ̅̅̅̅ 2 , highlighting a somehow slow convergence in the case  = 2. Faster convergence is achieved with  = 3 and 4, for which satisfactory results can be obtained with much less nodes N than  = 2. Simulations for  = 4 with N > 320,000 nodes were computationally too expensive and they have not been carried out.
Equal polynomial degrees  1 =  2 =  are employed for the Hermite-Chaos expansion over the M = 2 uncertain variables, requiring  + 1 = ( + 1) 2 deterministic RBF-FD solutions over the corresponding geometries.Based on the convergence study of subsection 4.1, N ≈ 200,000 nodes and  = 3 are employed as a reasonable balance between accuracy and computational effort for the deterministic solutions over the ( + 1) 2 required geometries.We point out that each of these deterministic solutions is obtained by generating a completely new meshless node distribution, requiring a post-processing mapping to the reference geometry with   = 0.
The convergence of mean and standard deviation of the average Nusselt numbers Nu ̅̅̅̅ 1 , Nu ̅̅̅̅ 2 and Nu ̅̅̅̅ (averaged Nusselt number over both wavy surfaces) for four different Hermite-Chaos polynomial degrees  = 1, 2, 3, 4 is reported in table 2 and table 3 for 5% and 10% standard deviation of the amplitudes, respectively.The mean values are almost exactly the same to the fourth digit, while a very fast convergence is achieved for the standard deviation  in both cases and for each Nu value, confirming the accuracy of the tensorial-expanded Hermite Chaos method.The computation of mean and standard deviation is then applied to the local Nusselt number over the wavy surfaces.The results for the case with 10% standard deviation of the amplitudes and  = 4 are shown in figure 3 for the internal and the external wavy surfaces.As expected, a moderate standard deviation of the local Nu number at the internal surface is found where the Nu value is high, but the highest values of the standard deviation are found at the external surface, especially in the upper part near  = 0 and  = 3.

Conclusions
In this work the Non-Intrusive Polynomial Chaos method is coupled to the RBF-FD meshless method for the accurate prediction of the uncertainties in laminar, incompressible and steady 3D thermo-fluid problems where the boundaries are described by stochastic variables.The main statistical properties  of any primitive or derived flow quantity can be accurately computed by using the minimum number of deterministic fluid-flow simulations required by the tensorial-expanded Hermite-Chaos formulation.This approach is employed to compute the statistical moments of both local and average Nusselt numbers in a forced convection problem over a 3D pipe with wavy surfaces with uncertain amplitude.
The main advantage of the presented meshless approach over classical mesh-based approaches is its capability to easily deal with complex 3D geometries, which is an additional advantage when coupling the RBF-FD method to the employed Non-Intrusive Polynomial Chaos method since multiple deterministic solutions over different geometries are required.Consistent results are found, suggesting that the presented approach can be used for the efficient quantification of the uncertainties in fluid flow problems on 3D geometries of practical interest.

Appendix A. Analytical solution in cylindrical coordinates
In the case  1 =  2 = 0, i.e., an annular section pipe where the internal and the external cylinders with nondimensional radii  1 and  2 are kept at temperatures  1 and  2 , respectively, a steady-state solution of equations ( 1

Figure 2 .
Figure 2. Node distribution with N ≈ 200,000 nodes (168,000 internal nodes, 32,000 boundary nodes): boundary nodes (a), enlarged view of a particular with internal nodes in red (b), computed temperature field on the boundary (c), enlarged view of a particular of the meshless temperature field with boundary nodes in red (d).

Figure 3 .
Figure 3. Mean and standard deviation of the local Nusselt number, stochastic case with 10% standard deviation of  1 and  2 .

Table 1 .
Average Nusselt numbers for node distributions with N nodes and RBF-FD polynomial degree P, deterministic case.

Table 2 .
Mean and standard deviation of the average Nusselt numbers, stochastic case with 5% standard deviation of  1 and  2 .

Table 3 .
Mean and standard deviation of the average Nusselt numbers, stochastic case with 10% standard deviation of  1 and  2 .