MULTIVARIATE INTERPOLATION USING POLYHARMONIC SPLINES

Data measuring and further processing is the fundamental activity in all branches of science and technology. Data interpolation has been an important part of computational mathematics for a long time. In the paper, we are concerned with the interpolation by polyharmonic splines in an arbitrary dimension. We show the connection of this interpolation with the interpolation by radial basis functions and the smooth interpolation by generating functions, which provide means for minimizing the L2 norm of chosen derivatives of the interpolant. This can be useful in 2D and 3D, e.g., in the construction of geographic information systems or computer aided geometric design. We prove the properties of the piecewise polyharmonic spline interpolant and present a simple 1D example to illustratethem.

Optimisation of microfluidic polymerase chain reaction devices

The invention and development of Polymerase Chain Reaction (PCR) technology have revolutionised molecular biology and molecular diagnostics. There is an urgent need to optimise the performance of these devices while reducing the total construction and operation costs. This study proposes a CFD-enabled optimisation methodology for continuous flow (CF) PCR devices with serpentine-channel structure, which enables the optimisation of DNA amplification efficiency and pressure drop to be explored while varying the width (W) and height (H) of the microfluidic (μ) channel. This is achieved by using a surrogate-enabled optimisation approach accounting for the geometrical features of a μCFPCR device by performing a series of simulations using COMSOL Multiphysics 5.4®. The values of the objectives are extracted from the CFD solutions, and the response surfaces are created using polyharmonic splines. Genetic algorithms are then used to locate the optimum design parameters. The results indicate that there is the possibility of improving the DNA concentration and the pressure drop in a PCR cycle by ~2.1 % ([W, H] = [400 μm, 50 μm]) and ~95.2 % ([W, H] = [400 μm, 80 μm]) respectively, by modifying its geometry.

A stabilized well-balanced RBF-FD meshless method for shallow-water equations

In this work, we are concerned with the study and computing of stabilized radial basis function-generated finite difference (RBF-FD) approximations for shallow-water equations. In order to obtain both stable and highly accurate numerical approximations of convection-dominated shallow-water equations, we use stabilized flat Gaussians (RBFSGA-FD) and polyharmonic splines with supplementary polynomials (RBFPHS-FD) as basis functions, combined with modified method of characteristics. These techniques are combined with careful design for the spatial derivative operators in the momentum flux equation, according to a general criterion for the exact preservation of the “lake at rest” solution in general mesh-based and meshless numerical schemes for the strong form of the shallow-water equations with bottom topography. Both structured and unsructured point clouds are employed for evaluating the influence of cloud refinement, size of local supports and maximal permissible degree of the polynomials in RBFPHS-FD.

Harmonic Convergence Estimation Through Strain Energy Superconvergence

Grid convergence in finite element analysis (FEA), despite a wide variety of tools available to date, remains an elusive and challenging task. Due to the complex and time-consuming process of remeshing and solving the finite element model (FEM), convergence studies can be a part of the most arduous portion of the modeling process and can even be impossible with FEMs unassociated with CAD. Existing a posteriori methods, such as relative error in the energy norm, provide a near arbitrary indication of the model convergence for eigenfrequencies. This paper proposes a new approach to evaluate the harmonic convergence of an existing model without conducting a convergence study. Strain energy superconvergence (SES) takes advantage of superconvergence points within a FEM and accurately recovers the strain energy within the model using polyharmonic splines, thus providing a more accurate estimate of the system's eigenfrequencies without modification of the FEM. Accurate eigenfrequencies are critical for designing for airfoil resonance avoidance and mistuned rotor response prediction. Traditional error estimation strategies fail to capture harmonic convergence as effectively as SES, potentially leading to a less accurate airfoil resonance and rotor mistuning prediction.