Order Interpolation Research Articles

Gaussian quadrature method with exponential fitting factor for two-parameter singularly perturbed parabolic problem

The parabolic convection-diffusion-reaction problem is examined in this work, where the diffusion and convection terms are multiplied by two small parameters, respectively. The proposed approach is based on a fitted operator finite difference method. The Crank-Nicolson method on uniform mesh is utilized to discretize the time variables in the first step. Two-point Gaussian quadrature rule is used for further discretizing these semi-discrete problems in space, and the second order interpolation of the first derivatives is utilized. The fitting factor’s value, which accounts for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second-order accurate and uniformly convergent. The proposed method’s applicability is validated by two examples, which yielded more accurate results than some other methods found in the literatures.

Stereo Bi-Telecentric Phase-Measuring Deflectometry.

Replacing the endocentric lenses in traditional Phase-Measuring Deflectometry (PMD) with bi-telecentric lenses can reduce the number of parameters to be optimized during the calibration process, which can effectively increase both measurement precision and efficiency. Consequently, the low distortion characteristics of bi-telecentric PMD contribute to improved measurement accuracy. However, the calibration of the extrinsic parameters of bi-telecentric lenses requires the help of a micro-positioning stage. Using a micro-positioning stage for the calibration of external parameters can result in an excessively cumbersome and time-consuming calibration process. Thus, this study proposes a holistic and flexible calibration solution for which only one flat mirror in three poses is needed. In order to obtain accurate measurement results, the calibration residuals are utilized to construct the inverse distortion map through bicubic Hermite interpolation in order to obtain accurate anchor positioning result. The calibrated stereo bi-telecentric PMD can achieve 3.5 μm (Peak-to-Valley value) accuracy within 100 mm (Width) × 100 mm (Height) × 200 mm (Depth) domain for various surfaces. This allows the obtaining of reliable measurement results without restricting the placement of the surface under test.

A Robust and Simple Method for Filling in Masked Data in Astronomical Images

Astronomical images often have regions with missing or unwanted information, such as bad pixels, bad columns, cosmic rays, masked objects, or residuals from imperfect model subtractions. In certain situations it can be essential, or preferable, to fill in these regions. Most existing methods use low order interpolations for this task. In this paper a method is described that uses the full information that is contained in the pixels just outside masked regions. These edge pixels are extrapolated inwards, using iterative median filtering. This leads to a smoothly varying spatial resolution within the filled-in regions, and ensures seamless transitions between masked pixels and good pixels. Gaps in continuous, narrow features can be reconstructed with high fidelity, even if they are large. The method is implemented in maskfill, an open-source MIT licensed Python package (https://github.com/dokkum/maskfill). Its performance is illustrated with several examples, and compared to several alternative interpolation schemes.

A large deformation multiphase continuum mechanics model for shock loading of soft porous materials

AbstractA large deformation, coupled finite‐element (FE) model is developed to simulate the multiphase response of soft porous materials subjected to high strain‐rate loading. The approach is based on the theory of porous media (TPM) at large deformations. Simplifications to the one‐dimensional regime studied in the numerical simulations follow. An overview of several different time integration schemes is presented for the purpose of solving the nonlinear dynamic coupled balance of momenta (mixture and fluid) and balance of mass of the mixture equations. Numerical examples are presented for (i) verification against closed‐form analytical solutions assuming small loads, (ii) demonstrating large deformation effects at high strain‐rate, and (iii) showing differences in deformations between a single‐phase elastodynamics model with occluded compressible pore fluid and a multiphase poroelastodynamics model at high strain‐rate. The multiphase model shows that the relative motion of the pore fluid significantly dampens the deformation response of the solid skeleton as compared to the single‐phase model, and makes it possible to extract quantitative values for the stresses of the different constituents, thereby allowing one to form preliminary conclusions about the onset of damage in the solid skeleton. The novelty of the current work is developing a multiphase, large deformation, mixture theory numerical model for high strain‐rate loading of soft porous materials. It was discovered that explicit, adaptive time‐stepping Runge–Kutta schemes offer high accuracy at relatively low cost when compared to traditional implicit or explicit central difference time‐stepping schemes for shock‐like loadings. Shock viscosity is added to the mixture momentum balance equation to regularize the shock front, and a stabilization term is added to the mixture mass balance equation to stabilize equal order interpolation finite elements for the coupled finite element solution of multiphase materials.

Study of the convergence of the Meshless Lattice Boltzmann Method in Taylor–Green, annular channel and a porous medium flows

The Meshless Lattice Boltzmann Method (MLBM) is a numerical tool that relieves the standard Lattice Boltzmann Method (LBM) from regular lattices and, at the same time, decouples space and velocity discretizations. In this study, we investigate the numerical convergence of MLBM in three benchmark tests: the Taylor–Green vortex, annular (bent) channel and a porous medium flow. We compare our MLBM results to LBM and to the analytical solution of the Navier–Stokes equation. We investigate the method’s convergence in terms of the discretization parameter, the interpolation order, and the LBM streaming distance refinement. We observe that MLBM outperforms LBM in terms of the error value for the same number of nodes discretizing the domain. We find that LBM errors at a given streaming distance δx and timestep length δt are the asymptotic lower bounds of MLBM errors with the same streaming distance and timestep length. Finally, we suggest an expression for the MLBM error that consists of the LBM error and other terms related to the semi-Lagrangian nature of the discussed method itself.

Mittag–Leffler expansions for inverse spectral problems with mixed data

We consider the inverse spectral problem for the Sturm–Liouville problem in the interval with the Dirichlet boundary conditions at both ends. We provide a method for the unique reconstruction of the potential that is known a priori on the subinterval , where m and r being positive integers and satisfying , by using the spectral data set consisting of all the eigenvalues and a subset of the norming constants. The method is based on the Mittag–Leffler decomposition which can help us to decompose an entire function of exponential type into two functions of smaller types. This decomposition allows the use of Lagrange interpolation in order to reconstruct the potential in the Sturm–Liouville problem. We also provide necessary and sufficient conditions related to the existence issues.

Multi-Effective Collocation Methods for Solving the Volterra Integral Equation with Highly Oscillatory Fourier Kernels

In this paper, we focus on the numerical solution of the second kind of Volterra integral equation with a highly oscillatory Fourier kernel. Based on the calculation of the modified moments, we propose four collocation methods to solve the equations: direct linear interpolation, direct higher order interpolation, direct Hermite interpolation and piecewise Hermite interpolation. These four methods are simple to construct and can quickly compute highly oscillatory integrals involving Fourier functions. We present the corresponding error analysis and it is easy to see that, in some cases, our proposed method has a fast convergence rate in solving such equations. In some cases, our proposed methods have significant advantages over the existing methods. Some numerical experiments demonstrating the efficiency of the four methods are also presented.

Investigations on adapted interpolation orders for a mixed isogeometric plate formulation

AbstractIn order to overcome locking effects that especially occur for lower order finite element formulations, different methods can be employed. This can be conducted using mixed formulations or adapted approximation orders, for instance. Hence, in order to tackle shear locking that is caused by non‐matching interpolation degrees in the shear strain equation, an irreducible and a mixed Reissner‐Mindlin plate formulation with accordingly adapted conforming discretizations are derived within the scope of this contribution. In addition, non‐uniform rational B‐splines (NURBS) are employed therefore, in order to benefit from the properties and refinement strategies offered by isogeometric analysis (IGA) and to achieve more accurate results. The effect of various combinations of interpolation orders on the convergence behavior and the ability to alleviate locking is investigated for both the irreducible and the mixed isogeometric plate formulation and examined for a benchmark example. This is also supplemented by investigations on the stability of the considered variants, tested by the existence of the correct number of zero‐energy modes.

Exposure-response relation for vibration-induced white finger: effect of different methods for predicting prevalence.

A pooled analysis of vibration-induced white finger (VWF) in population groups of workers has been performed using the results of a published meta-analysis as source material (Nilsson T, Wahlström J, Burström L. Hand-arm vibration and the risk of vascular and neurological diseases a systematic review and meta-analysis. PLoS One. 2017:12(7):e0180795. https://doi.org/10.1371/journal.pone.0180795). The methods of data selection follow those described previously by Scholz et al. (in Scholz MF, Brammer AJ, Marburg S. Exposure-response relation for vibration-induced white finger: inferences from a published meta-analysis of population groups. Int Arch Occup Environ Health. 2023a:96(5):757-770. https://doi.org/10.1007/s00420-023-01965-w) to enable comparison with the results of the present work. The analyzed epidemiologic studies contain different prevalences of VWF observed after different durations of employment involving exposure to the vibration of power tools and machines. These prevalences are transformed to 10% prevalence by either linear or polynomial (i.e. "S"-shaped curvilinear) interpolation in order to compare with the exposure-response relation contained in the relevant international standard (ISO 5349-1:2001). An exposure-response relation is constructed using regression analysis for the time (in years) to reach 10% prevalence in a population group, when subjected to a daily vibration exposure calculated according to the procedures specified in the standard, A(8). Good fits to the data are obtained when polynomial and linear prevalence interpolation is used. The 95-percentile confidence intervals (CIs) of the exposure-response relation predicted by polynomial prevalence interpolation lie at somewhat larger lifetime exposures than those obtained by linear prevalence interpolation. Uncertainty in the precision of polynomial prevalence interpolation is mitigated by giving equal weight to linear interpolation when interpreting the results. When the 95-percentile CIs of the exposure-response models obtained by linear and polynomial prevalence interpolation are used to define the most probable exposure-response relation, the resulting common range of values includes the ISO exposure-response relation. It is proposed that an exposure-response relation for the onset of VWF derived from a regression analysis is specified in terms of the lower limit of its CI. Hence, when exposure measures are constructed according to the ISO standard and equal weight is given to the results of the 2 methods for interpolating prevalence described here, the ISO exposure-response relation would be considered to provide a conservative estimate for a 10% prevalence of VWF to develop in a population group, at least for A(8) > 4 m/s2. It thus remains the relation to use for assessing exposure to hand-transmitted vibration in the workplace. Additional research is needed to resolve inconsistencies in the ISO method for calculating daily exposures.

Endpoint Geodesic Formulas on Graßmannians Applied to Interpolation Problems

Simple closed formulas for endpoint geodesics on Graßmann manifolds are presented. In addition to realizing the shortest distance between two points, geodesics are also essential tools to generate more sophisticated curves that solve higher order interpolation problems on manifolds. This will be illustrated with the geometric de Casteljau construction offering an excellent alternative to the variational approach which gives rise to Riemannian polynomials and splines.

Nonlinear Vibration Characteristics of Point Supported Isotropic and Symmetrically Laminated Plates

Nonlinear free and forced vibration characteristics of composite square plates either supported at four corner points or point supported at the middle of its edges as well as corners are investigated here using different high precision plate bending finite elements. It is observed that, higher order interpolation polynomial for the transverse displacement helps the plate bending finite elements to converge for the linear vibration frequencies. However, convergence for large amplitude vibration problems depends on the interpolation polynomials for in-plane displacements as well as transverse displacement. In general, the hardening nonlinearity (i.e., the increase of nonlinear frequency with vibration amplitude) of point supported plates is less than those of immovable simply supported plates.

RAM: Rapid Advection Algorithm on Arbitrary Meshes

The study of many astrophysical flows requires computational algorithms that can capture high Mach number flows, while resolving a large dynamic range in spatial and density scales. In this paper we present a novel method, RAM: Rapid Advection Algorithm on Arbitrary Meshes. RAM is a time-explicit method to solve the advection equation in problems with large bulk velocity on arbitrary computational grids. In comparison with standard upwind algorithms, RAM enables advection with larger time steps and lower truncation errors. Our method is based on the operator splitting technique and conservative interpolation. Depending on the bulk velocity and resolution, RAM can decrease the numerical cost of hydrodynamics by more than one order of magnitude. To quantify the truncation errors and speed-up with RAM, we perform one- and two-dimensional hydrodynamics tests. We find that the order of our method is given by the order of the conservative interpolation and that the effective speed-up is in agreement with the relative increment in time step. RAM will be especially useful for numerical studies of disk−satellite interaction, characterized by high bulk orbital velocities and nontrivial geometries. Our method dramatically lowers the computational cost of simulations that simultaneously resolve the global disk and potential well inside the Hill radius of the secondary companion.

Stochastic spectral cell method for structural dynamics and wave propagations

AbstractThe spectral element method when enriched with the concept of fictitious domain constitutes the spectral cell method which is appropriately able to model geometrically complicated problems by simplifying the mesh generation. In this paper, we propose a new computational framework called stochastic spectral cell method (SSCM) as a stochastic development of the spectral cell method for uncertainty quantification applied to structural dynamics and wave propagations. The stochastic form of spectral cell method has not been developed yet. Hence, the SSCM is a novel stochastic method comprising the advantages of spectral cell method for computational mechanics considering the geometrical complexity resulting from computer‐aided design. The SSCM is established by incorporating the polynomial chaos and Karhunen‐Loève expansions for the uncertainty quantification, and the interpolation functions of spectral elements for the spatial discretization. In the SSCM, the numerical solution of the Fredholm integral equation of the second kind required in Karhunen‐Loève expansion is obtained using the spectral cell method. The use of Cartesian mesh, diagonal mass matrix, and higher order interpolation functions of spectral elements for stochastic dynamic analysis lead to desirable computational efficiency and accuracy of the SSCM. The capability, effectiveness, and accuracy of the proposed SSCM are demonstrated with several numerical examples of wave propagation as well as dynamic analysis of structures.

An MP-DWR method for h-adaptive finite element methods

In a dual-weighted residual method based on the finite element framework, the Galerkin orthogonality is an issue that prevents solving the dual equation in the same space as the one for the primal equation. In the literature, there have been two popular approaches to constructing a new space for the dual problem, i.e., refining mesh grids (h-approach) and raising the order of approximate polynomials (p-approach). In this paper, a novel approach is proposed for the purpose based on the multiple-precision technique, i.e., the construction of the new finite element space is based on the same configuration as the one for the primal equation, except for the precision in calculations. The feasibility of such a new approach is discussed in detail in the paper. In numerical experiments, the proposed approach can be realized conveniently with C++ template. Moreover, the new approach shows remarkable improvements in both efficiency and storage compared with the h-approach and the p-approach. It is worth mentioning that the performance of our approach is comparable with the one through a higher order interpolation (i-approach) in the literature. The combination of these two approaches is believed to further enhance the efficiency of the dual-weighted residual method.

Complex-variable, high-precision formulation of the consistent boundary element method for 2D potential and elasticity problems

The collocation boundary element method was recently entirely revisited on the basis of a consistent derivation of Somigliana’s identity in terms of weighted residuals. Both conceptually and for the sake of code implementation, the correct traction force interpolation along generally curved boundaries, as for elasticity problems, leads to the enunciation of an inedited, actually long-sought, convergence theorem as well as to considerable numerical simplifications. Numerical evaluations for two-dimensional problems require exclusively Gauss–Legendre quadrature plus eventual correction terms obtained analytically regardless of the order or shape of the implemented boundary element interpolation. Arbitrarily high precision and accuracy is achievable for low-cost computation, as eventual mesh-subdivision refinements should take place only if the mechanical simulation demands — and not just for numerical evaluations. We now show that considerable simplification is obtained by switching the formulation from real to complex variable. Precision, round-off errors, and accuracy of a given numerical implementation may be kept – identifiably and separately – under control, as assessed for some potential and elasticity examples with extremely challenging topologies. In fact, source-field distances may be arbitrarily small — far smaller than deemed feasible in continuum mechanics, as we resort to nothing else than the problem’s mathematics.

Convergence analysis of pressure reconstruction methods from discrete velocities

Magnetic resonance imaging allows the measurement of the three-dimensional velocity field in blood flows. Therefore, several methods have been proposed to reconstruct the pressure field from such measurements using the incompressible Navier–Stokes equations, thereby avoiding the use of invasive technologies. However, those measurements are obtained at limited spatial resolution given by the voxel sizes in the image. In this paper, we propose a strategy for the convergence analysis of state-of-the-art pressure reconstruction methods. The methods analyzed are the so-called Pressure Poisson Estimator (PPE) and Stokes Estimator (STE). In both methods, the right-hand side corresponds to the terms that involving the field velocity in the Navier–Stokes equations, with a piecewise linear interpolation of the exact velocity. In the theoretical error analysis, we show that many terms of different order of convergence appear. These are certainly dominated by the lowest-order term, which in most cases stems from the interpolation of the velocity field. However, the numerical results in academic examples indicate that only the PPE may profit of increasing the polynomial order, and that the STE presents a higher accuracy than the PPE, but the interpolation order of the velocity field always prevails. Furthermore, we compare the pressure estimation methods on real MRI data, assessing the impact of different spatial resolutions and polynomial degrees on each method. Here, the results are consistent with the academic test cases in terms of sensitivity to polynomial order as well as the STE showing to be potentially more accurate when compared to reference pressure measurements.

Effective collocation methods to solve Volterra integral equations with weakly singular highly oscillatory Fourier or Airy kernels

In this paper two collocation methods, the direct linear interpolation collocation method and the direct high order interpolation collocation method, are proposed to solve the second kind of Volterra integral equations with weakly singular highly oscillatory Fourier or Airy kernels. The purpose is to solve the equations by Kummer hypergeometric and Gamma functions for solving the modified moments, where the modified moments are transformed from the integration interval [0,x] to the interval [−1,1]. The detailed collocation procedure is provided and the effectiveness of these algorithms are proved by convergence analysis and numerical experiments. Further, numerical examples are used to prove that the direct linear interpolation collocation method and the direct high order interpolation collocation method are also effective for solving highly oscillatory equations without weakly singularities.

A hybrid implicit-explicit discontinuous Galerkin spectral element time domain (DG-SETD) method for computational elastodynamics

SUMMARY A hybrid implicit-explicit (IMEX) discontinuous Galerkin spectral element time domain (DG-SETD) algorithm is proposed to simulate 3D elastic wave propagation in inhomogeneous media. In this method, the original problem can be divided into a number of well designed subdomains, and the mesh generation of different subdomains is completely independent, thus allowing flexible spatial discretization of complex geometry. The neighboring subdomains are connected by a Riemann transmission condition (RTC), and spectral elements with different interpolation orders can be used in different subdomains to maximize the computational efficiency of multiscale problems to facilitate parallel computing for different subdomains. In particular, the explicit or implicit time iteration scheme can be appropriately selected for a subdomain according to the size of its mesh elements to increase the global time step increment, thus giving higher computational efficiency: For subdomains with coarse meshes, the explicit time integration scheme is used and the time step increment is limited by the Courant−Friedrichs−Lewy (CFL) stability condition; for subdomain with fine structures and thus fine meshes, an implicit time integration scheme is used so that a large time step increment can be used without affecting the stability. In addition, a jump condition of displacement and velocity is introduced to accurately simulating fractures and faults, including lossless and viscous fractures with plane, curve or cross structures. This avoids the volume modeling of the extremely thin fracture structures, and effectively reduces the number of degrees of freedoms (DoFs) in the discretized system without the loss of accuracy. The accuracy, robustness and efficiency of the DG-SETD algorithm are demonstrated by quantitative comparisons of the waveforms with the commercial finite element software COMSOL.

Accurate coarse mesh simulation of delamination in composites using a novel hp-adaptive cohesive element

Cohesive elements are routinely used to simulate delamination between plies in composites. However, the allowable maximum size of these elements is limited, owing to the presence of a very small process zone ahead of the cohesive crack-tip, which needs to be discretised with multiple elements for obtaining an accurate failure response. This restricts their applicability beyond analysing delamination in coupon scale problems, due to the need for a very fine mesh and associated large problem size. In this paper, a novel hp cohesive element is developed to address this problem. This element is capable of simultaneous mesh refinement and elevation of interpolation order, locally and adaptively around the propagating cohesive crack tip. Additional degrees of freedom associated with the hp refinement are handled by extra nodes already supplied in the initial mesh file, that are selectively activated by the program as required. Several example problems are presented showing the ability of this method to improve the prediction of delamination onset and growth in extremely coarse meshes with a reduced computational time.

Two-scale modeling of particle crack initiation and propagation in particle-reinforced composites by using microscale analysis based on Voronoi cell finite element model

In this study, a two-scale finite element model has been developed for the elastic analysis of composite materials. The coarse finite element mesh was assumed for the macroscale analysis, while Voronoi cell finite element model (VCFEM) was used for the microscale analysis of heterogeneous domains. The adaptive coupling between the two scales in non-critical and critical regions was accomplished through the master–slave nodes method. Two scale elements were connected together through the use of shared nodes and the order of the edge displacement interpolation. A new extended VCFEM was proposed to simulate particle damage generation, and a remeshing algorithm was constructed to simulate particle crack generation and propagation in particle-reinforced composites. For crack propagation analysis, four new combined Voronoi elements were developed and analyzed at the microscale. The obtained results for the stresses, stress intensity factors, and crack path were compared with those calculated using the commercial calculation software Abaqus. The two sets of results are in very good agreement, confirming the accuracy of the proposed method used for determining the stress concentrations and also for coarse-mesh discretization.