High-order Interpolation Research Articles

An efficient hybrid multi-resolution WCNS scheme for solving compressible flows

A hybrid finite difference fifth-order multi-resolution weighted compact nonlinear scheme (WCNS) is presented for simulating the inviscid and viscous compressible flows in this paper. This scheme borrows and improves the idea of the recently developed multi-resolution WENO scheme for node-to-midpoint nonlinear interpolation procedures of the WCNS scheme, which implements higher-order approximate interpolation at midpoints based on a series of nested stencils. This strategy can achieve the expected design accuracy under different linear weights, and can gradually decrease from high-order accuracy to first-order accuracy near the discontinuities. In addition, a hybridization strategy is proposed to further improve the computational efficiency of the presented multi-resolution WCNS scheme. The strategy does not depend on a specific problem to be solved, nor does it contain any problem-related artificial parameters. Finally, some benchmark examples are used to verify the performance of the presented scheme in terms of numerical accuracy, shock capture capability, robustness, computational efficiency, etc.

Finite element mesh refinement for in-plane and out-of-plane vibration of variable geometrical Timoshenko beams based on superconvergent vibration modes

PurposeIn this paper, a superconvergent patch recovery method is proposed for superconvergent solutions of modes in the finite element post-processing stage of variable geometrical Timoshenko beams. The proposed superconvergent patch recovery method improves the solution speed and accuracy of the finite element analysis of a curved beam. The free vibration and natural frequency of the beam were considered for studying forced vibrations and structural resonance. Beam vibration mode analysis was performed for high-precision vibration mode solutions and frequency values. The proposed method can be used to compute beam vibration modes of beams with different shapes and boundary conditions as well as variable cross sections and curvatures. The purpose of this paper is to address these issues.Design/methodology/approachAn adaptive method was proposed to analyse the in-plane and out-of-plane free vibrations of the variable geometrical Timoshenko beams. In the post-processing stage of the displacement-based finite element method, the superconvergent patch recovery method and high-order shape function interpolation technique were used to obtain the superconvergent solution of mode (displacement). The superconvergent solution of mode was used to estimate the error of the finite element solution of mode in the energy form under the current mesh. Furthermore, an adaptive mesh refinement was proposed by mesh subdivision to derive an optimised mesh and accurate finite element solution to meet the preset error tolerance.FindingsThe results computed using the proposed algorithm were in good agreement with those computed using other high-precision algorithms, thus validating the accuracy of the proposed algorithm for beam analysis. The numerical analysis of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams helped verify that the solutions of frequencies were consistent with the results obtained using other specially developed methods. The proposed method is well suited for the mesh refinement analysis of a curved beam structure for analysing the changes in high-order vibration mode. The parts where the vibration mode changed significantly were locally densified; a relatively fine mesh division was adopted that validated the reliability of the mesh optimisation processing of the proposed algorithm.Originality/valueThe proposed algorithm can obtain high-precision vibration solutions of variable geometrical Timoshenko beams based on more optimized and reasonable meshes than the conventional finite element method. Furthermore, it can be used for vibration problems of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams. The proposed algorithm can be extended for application in superconvergent computation and adaptive analysis of finite element solutions of general structures and solid deformation fields and used for adaptive analysis of more complex plates, shells and three-dimensional structures. Additionally, this method can analyse the vibration and stability of curved members with crack damage to obtain high-precision vibration modes and instability modes under damage defects.

Implicit large-eddy simulation of turbulent plane jet at Re = 104

In this study, implicit large-eddy simulations (ILESs) are performed for a turbulent plane jet at a Reynolds number 104 to assess the capability of ILESs in simulating turbulent transport in free shear flows. To investigate the effect of resolved turbulent scales on the self-similar behavior of the plane jet, ILESs are conducted using fifth- (ILES+P5), seventh- (ILES+P7), and ninth-order polynomial (ILES+P9) interpolation methods, as well as the third-order MUSCL method (ILES+M3). ILES+P5, ILES+P7, and ILES+P9 reproduced the results of the direct numerical simulation (DNS) for a plane jet at Re = 104. The resolution of the power spectral density of ILESs with high-order interpolation schemes, which agrees well with the DNS, increases as the order of spatial accuracy increases and resolves more than 80% of the total energy. Hence, we conclude that the spatial resolution to resolve up to the frequency region that contains approximately 80% of the turbulent energy is required to obtain good agreement with the results of DNS, which is widely accepted in LES with the subgrid-scale (SGS) model and is also valid for ILES.

Simulation of non-linear structural elastodynamic and impact problems using minimum energy high-order bases

We present the application of simultaneous diagonalization and minimum energy (SDME) high-order finite element modal bases for simulation of transient non-linear elastodynamic problem, including impact cases with Hookean and neo-Hookean hyperelastic materials. The bases are constructed using procedures for simultaneous diagonalization of the internal modes and Schur complement of the boundary modes from the standard nodal and modal bases, constructed using Lagrange and Jacobi polynomials, respectively. The implementation of these bases in a high-order finite element code is straightforward, since the procedure is applied only to the one-dimensional expansion bases. Non-linear transient structural problems with large deformation , hyperelastic materials and impact are solved using the obtained bases with explicit and implicit time integration procedures. Iterative solutions based on preconditioned conjugate gradient methods are considered. The performance of the proposed bases in terms of the number of iterations of pre-conditioned conjugate gradient methods and computational time are compared with the standard nodal and modal bases. The SDME bases are accurate and provide better conditioned equations for iterative solutions in general, leading to substantial reduction in processing time over an order of magnitude compared to other modal bases. Our numerical tests obtained speedups up to 41 using the considered bases when compared to the standard modal basis. Results for a two-dimensional impact problem approximated with the SDME bases showed that the implicit Newmark method performed much better in terms of processing time and contact stress error when compared to the standard explicit central difference method with lumped mass matrix . • Application of simultaneous diagonalization procedure and minimum energy highorder bases to non-linear structural and impact transient problems. • Simple modifications of high-order bases improve conditioning of system of equations obtained from the high-order approximation of the considered problems, with remarkable gains in performance of iterative solvers. • Light variation on the number of pre-conditioned gradient iterations in terms of the polynomial order. • Speed-ups over 40 when compared to the standard modal Jacobi high-order bases used in high-order interpolation.

On the performance of a Chimera-FEM implementation to treat moving heat sources and moving boundaries in time-dependent problems

Problems with moving sources and moving inner boundaries in transient regime are of high interest in many research fields and engineering applications. One approach to properly tackle such problems is based on the Chimera method for non-matching grids, where each moving object is defined on a fine mesh that moves across the fixed coarse background. In this way, the high gradients around moving sources or boundaries are captured by the fine mesh without need of globally fine fixed meshes or adaptive refinement. In this work, Chimera is implemented in the framework of the finite element method, following an arbitrary Lagrangian–Eulerian formulation for the moving meshes and a standard Eulerian formulation for the fixed background mesh. Further, in case of convection-dominated problems, the scheme is stabilized using the streamline upwind Petrov–Galerkin method. The coupling between the moving fine meshes and the coarse fixed one is achieved via Dirichlet boundary conditions and a high-order interpolation algorithm. The performance of the proposed methodology in terms of accuracy and stability is assessed by means of numerical tests to be compared with equivalent problems solved using fixed meshes. Further tests serve to highlight the good performance of the proposed Chimera-based finite element method to address both convection-dominated problems with multiple moving boundaries and sources, and three-dimensional arc welding processes. • An algebraic-based Chimera-FEM scheme for unsteady problems is proposed. • Remeshing is avoided for large movements of heat sources and boundaries. • Advective-dominant problems can be addressed with Chimera-FEM and SUPG. • Good accuracy is obtained compared to conventional one-domain approaches. • The scheme is suitable to be used on 3D welding process simulations.

Towards nonuniform distributions of unisolvent weights for high-order Whitney edge elements

We propose to extend results on the interpolation theory for scalar functions to the case of differential k-forms. More precisely, we consider the interpolation of fields in {mathcal P}^-_r varLambda ^k(T), the finite element spaces of trimmed polynomial k-forms of arbitrary degree r ge 1, from their weights, namely their integrals on k-chains. These integrals have a clear physical interpretation, such as circulations along curves, fluxes across surfaces, densities in volumes, depending on the value of k. In this work, for k=1, we rely on the flexibility of the weights with respect to their geometrical support, to study different sets of 1-chains in T for a high order interpolation of differential 1-forms, constructed starting from “good” sets of nodes for a high order multi-variate polynomial representation of scalar fields, namely 0-forms. We analyse the growth of the generalized Lebesgue constant with the degree r and preliminary numerical results for edge elements support the nonuniform choice, in agreement with the well-known nodal case.

Spectral element method for 3-D controlled-source electromagnetic forward modelling using unstructured hexahedral meshes

SUMMARYFor forward modelling of realistic 3-D land-based controlled-source electromagnetic (EM) problems, we develop a parallel spectral element approach, blending the flexibility and versatility of the finite element method in using unstructured grids with the accuracy of the spectral method. Complex-shaped structures and topography are accommodated by using unstructured hexahedral meshes, in which the elements can have curved edges and non-planar faces. Our code is the first spectral element algorithm in EM geophysics that uses the total field formulation (here that of the electric field). Combining unstructured grids and a total field formulation provides advantages in dealing with topography, in particular, when the transmitter is located on rough surface topography. As a further improvement over existing spectral element methods, our approach does not only allow for arbitrary distributions of conductivity, but also of magnetic permeability and dielectric permittivity. The total electric field on the elements is expanded in terms of high-order Lagrangian interpolants, and element-wise integration in the weak form of the boundary value problem is accomplished by Gauss–Legendre–Lobatto quadrature. The resulting complex-valued linear system of equations is solved using the direct solver MUMPS, and, subsequently, the magnetic field is computed at the points of interest by Faraday’s law. Five numerical examples comprehensively study the benefits of this algorithm. Comparisons to semi-analytical and finite element results confirm accurate representation of the EM responses and indicate low dependency on mesh discretization for the spectral element method. A convergence study illuminates the relation between high order polynomial approximation and mesh size and their effects on accuracy and computational cost revealing that high-order approximation yields accurate modelling results for very coarse meshes but is accompanied by high computational cost. The presented numerical experiments give evidence that 2nd and 3rd degree polynomials in combination with moderately discretized meshes provide better trade-offs in terms of computational resources and accuracy than lowest and higher order spectral element methods. To our knowledge, our final example that includes pronounced surface topography and two geometrically complicated conductive anomalies represents the first successful attempt at using 2nd order hexahedral elements supporting curved edges and non-planar faces in controlled-source EM geophysics.

ENO-based high-order data-bounded and constrained positivity-preserving interpolation

A number of key scientific computing applications that are based upon tensor-product grid constructions, such as numerical weather prediction (NWP) and combustion simulations, require property-preserving interpolation. Essentially non-oscillatory (ENO) interpolation is a classic example of such interpolation schemes. In the aforementioned application areas, property preservation often manifests itself as a requirement for either data boundedness or positivity preservation. For example, in NWP, one may have to interpolate between the grid on which the dynamics is calculated to a grid on which the physics is calculated (and back). Interpolating density or other key physical quantities without accounting for property preservation may lead to negative values that are nonphysical and result in inaccurate representations and/or interpretations of the physical data. Property-preserving interpolation is straightforward when used in the context of low-order numerical simulation methods. High-order property-preserving interpolation is, however, nontrivial, especially in the case where the interpolation points are not equispaced. In this paper, we demonstrate that it is possible to construct high-order interpolation methods that ensure either data boundedness or constrained positivity preservation. A novel feature of the algorithm is that the positivity-preserving interpolant is constrained; that is, the amount by which it exceeds the data values may be strictly controlled. The algorithm we have developed comes with theoretical estimates that provide sufficient conditions for data boundedness and constrained positivity preservation. We demonstrate the application of our algorithm on a collection of 1D and 2D numerical examples, and show that in all cases property preservation is respected.

A hybrid smoothed particle hydrodynamics coupled to a fictitious domain method for particulate flows and its application in a three-dimensional printing process

A hybrid smoothed particle hydrodynamics (SPH) coupled to a direct-forcing fictitious domain (FD) scheme is proposed for the simulation of particulate flows. The new method is effective in calculating the particle-fluid interaction forces and uses a kernel interpolation function to realize the momentum exchange between fluids and particles. A higher-order interpolation scheme is applied to preserve the consistency and accuracy of the kernel function. We first test our method by simulating the benchmark of flow past a cylinder at Reynolds number ranging from 20 to 200 and obtain the convergent results, including accurate lift and drag coefficients, Strouhal number, length of the recirculation bubble, and separation angle. Subsequently, we verify our new method for the typical particulate flows, including the motion of a neutrally-buoyant cylinder in Couette flow, the motion of a neutrally-buoyant cylinder in a planar Poiseuille flow, the sedimentation of a circular cylinder in a channel, and the motion of a neutrally-buoyant slender particle in Couette flow. A good agreement with the existing experimental data, with other numerical results, and with available theoretical solutions is obtained indicating the accuracy and the robustness of the new method. Subsequently, our SPH-FD is applied to simulate a slender fiber in a three-dimensional (3D) printing process, showing a good potential and advantages in simulating particulate flows of some industrial processes.

M-VCUT level set method for the layout and shape optimization of stiffeners in plate

Stiffened plates are widely used in engineering due to their relatively high rigidly-to-weight ratio. The stiffener layout and shape can significantly affect the mechanical properties of the plate. In this paper, a novel approach is developed in the framework of the Multiple Variable Cutting (M-VCUT) level set method for optimizing the layout and shape of stiffeners based on the Mindlin plate theory. In this approach, the layout and shape of stiffener can be controlled by adjusting the lower and upper limits of the design variables (cutting heights of the cutting surfaces at the cell nodes) conveniently. The coordinate mapping technique from the rectangular cell to the quadrilateral cell enables the shape of design domain to be arbitrary. In addition, multi-node cell with high-order polynomial interpolation technique is also applied for generating more complex geometry of stiffener. The numerical examples meshed by rectangular and quadrilateral cells are conducted for verifying the effectiveness of the proposed approach.

An Overview of Higher-Order Beam Elements Based on the Absolute Nodal Coordinate Formulation

Abstract In this paper, beam elements with particular emphasis on higher-order elements based on the absolute nodal coordinate formulation (ANCF) are thoroughly investigated from the perspective of interpolation procedure and numerical performance. A straightforward and modularized procedure to construct the shape function is proposed. Based on the unified shape function formulation, the research examines how axial and transverse interpolation strategies impact element performance. Two beams in the pure bending scenario are analyzed. The comparison study reveals that higher-order interpolation in the axial and transverse directions is necessary to represent the highly curved deformation modes and alleviate Poisson locking. The Princeton beam and a thicker beam are then studied to assess the accuracy, convergence, and numerical stability of different beam elements. Conclusions are: (1) Higher-order beam elements are generally more accurate but converge more slowly. (2) To guarantee high accuracy, a complete set of transverse quadratic gradients must be adopted in the quadratic elements, and a higher-order transverse interpolation is necessary to capture the warping effect. (3) To avoid slow convergence, the axial order should not be lower than the transverse order. (4) Higher-order beam elements lead to a stiffness matrix with a larger condition number. With an inappropriate length to cross section ratio, the transverse cubic element results in an ill-conditioned stiffness matrix that brings numerical instability. (5) The numerical stability of higher-order beam elements are more sensitive to the length to cross section ratio of the meshed beam.

Analysis/application of stabilization by the over-integration technique in CBS-SEM for incompressible flow

When the characteristic-based split (CBS) scheme is combined with spectral element method (SEM), high-order nonlinear terms in CBS scheme produce non-negligible aliasing error, which causes severe stability problems. In this paper, a stabilized framework based on CBS-SEM for incompressible flow simulation is established with stabilization by over-integration. The introduced over-integration addresses the aliasing-driven instability by approximating nonlinear terms with higher-order interpolations. A numerical experiment is presented to assess the performance of the stabilized framework in terms of the stability range, solution quality, and computational efficiency. The main conclusions are as follows: (1) Over-integration effectively extends the range of stability for CBS-SEM, where the increase in low-order interpolations is more significant. (2) For extremely under-resolved problems, two types of instabilities, including instability caused by the limitation of CFL condition and instability triggered by aliasing error, are observed in the numerical experiment. The stabilized framework is verified to be effective in handling both types of instability. (3) The order of over-integration only slightly affects the stability, but it markedly improves the capture of complex flow structure for high-order interpolations. In addition, a numerical investigation on incompressible flow past an airfoil is performed to further demonstrate the effectiveness of the framework.

Two-Dimensional Fourier Continuation and Applications

This paper presents a method (2D-FC) for construction of bi-periodic extensions of smooth non-periodic functions defined over general two-dimensional smooth domains. The approach can be directly generalized to domains of any given dimensionality, and even to non-smooth domains, but such generalizations are not considered here. The 2D-FC extensions are produced in a two-step procedure. In the first step the one-dimensional Continuation method is applied along a discrete set of outward boundary-normal directions to produce, along such directions, continuations that vanish outside a narrow interval beyond the boundary. Thus, the first step of the algorithm produces blending-to-zero along normals for the given function values. In the second step, the extended function values are evaluated on an underlying Cartesian grid by means of an efficient, high-order boundary-normal interpolation scheme. A Continuation expansion of the given function can then be obtained by a direct application of the two-dimensional FFT algorithm. Algorithms of arbitrarily high order of accuracy can be obtained by this method. The usefulness and performance of the proposed two-dimensional Continuation method are illustrated with applications to the Poisson equation and the time-domain wave equation within a bounded domain. As part of these examples the novel Fourier Forwarding solver is introduced which, propagating plane waves as they would in free space and relying on certain boundary corrections, can solve the time-domain wave equation and other hyperbolic partial differential equations within general domains at computing costs that grow sublinearly with the size of the spatial discretization.

Attitude Reconstruction From Inertial Measurement: Mitigating Runge Effect for Dynamic Applications

Time-equispaced inertial measurements are practically used as inputs for motion determination. Polynomial interpolation is a common technique of recovering the gyroscope signal but is subject to a fundamentally numerical stability problem due to the Runge effect on equispaced samples. This paper reviews the theoretical results of Runge phenomenon in related areas and proposes a straightforward borrowing-and-cutting (BAC) strategy to depress it. It employs the neighboring samples for higher-order polynomial interpolation but only uses the middle polynomial segment in the actual time interval. The BAC strategy has been incorporated into attitude computation by functional iteration, leading to accuracy benefit of several orders of magnitude under the classical coning motion. It would potentially bring significant benefits to the inertial navigation computation under sustained dynamic motions.

A hybrid spectral element-infinite element approach for 3D controlled-source electromagnetic modeling

In this study, we developed a novel hybrid approach for 3D controlled-source electromagnetic (CSEM) forward modeling. The hybrid approach combines vector spectral element method (SEM) with scalar infinite element method (IFEM) by using infinite element instead of traditional truncated boundary. Based on Maxwell's equations, the curl-curl control equation satisfied by secondary electric field is derived. The boundary value problem in differential form is transformed into integral weak form by using Galerkin weighted residual method. Then the global problem is discretized by element subdivision and high-order orthogonal basis function interpolation. The computational domain is discretized using hexahedral spectral elements and a single layer of infinite elements is added outside the domain. The SEM with high-order Gauss-Lobatto-Legendre (GLL) vector basis functions is used to represent the interior electric field, and the IFEM with high-order Gauss-Lobatto-Legendre (GLL) scalar basis functions is uesd as a boundary condition. Finally, the global numerical solution is obtained by solving large-scale linear equations. The layered model verifies the accuracy and correctness of the approach, and the 3D model verifies the efficiency and numerical stability of the approach. Numerical experiments show that the hybrid SEM-IFEM approach developed in this study can reach similar accuracy to the conventional spectral element method, but the computational domain required can be much smaller, and the corresponding CPU time and memory requirement are greatly reduced, namely, the high precision three-dimensional electromagnetic response can be obtained with less computational resource consumption.

A nonlinear compressible flow disturbance formulation for adaptive mesh refinement wavepacket tracking in hypersonic boundary-layer flows

A new numerical approach for simulating nonlinear wavepackets in hypersonic boundary-layers is presented. The adaptive mesh refinement wavepacket tracking (AMR-WPT) method has been developed as an efficient alternative to conventional direct numerical simulations (DNS). The AMR-WPT method employs the nonlinear disturbances equations (NLDE), an overset dual mesh approach with higher-order interpolation, and adaptive mesh refinement (AMR) to track wavepackets in hypersonic boundary-layer flows. After introducing the numerical details, the method is used to simulate linear and nonlinear wavepackets for an axisymmetric M=9.81 straight cone and 2-D/3-D M=5.35 flat plate boundary-layer. The simulation results are compared against classical stability and transition prediction tools, such as linear stability theory (LST), parabolized stability equations (PSE) and DNS. It is demonstrated that the AMR-WPT method requires only about 10% of the number of grid points when compared to DNS of a nonlinear wavepacket inside a hypersonic flat plate boundary-layer flow.

A novel combination of mathematical modelling approaches for simulating nearly hypersonic, viscous, reacting magnetogasdynamic flows

This work focuses on the mathematical combination of magnetogasdynamics (MGD) and the Dunn and Kang chemical kinetic model with a multi-component Harten–Lax–van Leer Contact (HLLC) local Riemann solver using high-order interpolation schemes. Due to the nature of nearly hypersonic, unsteady, viscous, reacting MGD flows, the gas dissociates which is taken into account with 11 species and 26 chemical reactions along with the temperature dependent transport properties through the Wilke model. In the present work, the electromagnetic effects are considered by solving a Poisson equation simultaneously with the chemical reaction equations. The non-linear convective terms are treated numerically with third-, fifth- and seventh-order Weighted Essentially Non-Oscillatory (WENO) interpolation schemes along with second-, fourth- and sixth-order central finite difference schemes for the viscous terms. The combination of these numerical and physical models are analysed for stability and validated for benchmark problems such as (a) Hartmann flow as MGD validation test case and (b) the shock-tube problem for testing the chemical reaction model in which case the reaction rate controlling temperature suggested by Park has also been taken into account. Furthermore, a complex physical problem of an external aerodynamic flow over the cube for the Mach number varying from 1.25 to 3 have also been investigated, because a limited number of studies is available for understanding the flow behaviour of sharp cornered blunt bodies such as cubes. The knowledge of shock structures of these objects has become imperative to predict the atmospheric re-entry trajectories of de-orbiting CubeSats, space debris or fragments of launch vehicles. The main physical finding is that the shock structures are observed to be much more resolved compared to previous works, especially in the wake region. The knowledge gap is bridged by detailing the shock structures and showing the dependence of the shock standoff distance on total enthalpy of the flow.

A semi-Lagrangian meshfree Galerkin method for convection-dominated partial differential equations

A new numerical method, which is based on the coupling between semi-Lagrangian (SL) method and element free Galerkin (EFG) method, is developed for convection–diffusion partial differential equations with dominated convection terms. It is the first time that the SL method combined with the EFG method to solve the transient convection–diffusion equations. This combination has been more effective than expected. Compared with the existing SL methods, the proposed method shows several new advantages. Firstly, it takes full advantages of meshfree methods, therefore, the implementation difficulties of the SL method in backward tracing and departure point interpolation are greatly reduced. As interpolating the function values to a departure point, complex searching algorithm is not needed to determine which grid cell contains the departure point, and higher order interpolation is not essential to ensure the global accuracy. Meanwhile, in addition to having good stability and allowing the use of large time-step sizes, the new method realizes the continuous transition from convection–diffusion equation to pure convection equation and provides a uniform numerical format for these two kinds of equations. A problem with available analytical solution is solved to analyze the convergence behavior of the proposed method. It is demonstrated that the method can achieve the optimal convergence rate. Subsequently, the method is applied to solve the 1D and 2D Burgers’ equations. The results show that the proposed method can indeed obtain surprisingly good results for the Burgers’ equations in the very challenging convection-dominated and pure convection cases. These results are better than almost all the ones we have seen in literature. The good performance of this method in solving the challenging Burgers’ equations makes us believe that it will be a very promising method for solving convection-dominated problems.

An hp-version adaptive finite element algorithm for eigensolutions of moderately thick circular cylindrical shells via error homogenisation and higher-order interpolation

PurposeThis study presents a novel hp-version adaptive finite element method (FEM) to investigate the high-precision eigensolutions of the free vibration of moderately thick circular cylindrical shells, involving the issues of variable geometrical factors, such as the thickness, circumferential wave number, radius and length.Design/methodology/approachAn hp-version adaptive finite element (FE) algorithm is proposed for determining the eigensolutions of the free vibration of moderately thick circular cylindrical shells via error homogenisation and higher-order interpolation. This algorithm first develops the established h-version mesh refinement method for detecting the non-uniform distributed optimised meshes, where the error estimation and element subdivision approaches based on the superconvergent patch recovery displacement method are introduced to obtain high-precision solutions. The errors in the vibration mode solutions in the global space domain are homogenised and approximately the same. Subsequently, on the refined meshes, the algorithm uses higher-order shape functions for the interpolation of trial displacement functions to reduce the errors quickly, until the solution meets a pre-specified error tolerance condition. In this algorithm, the non-uniform mesh generation and higher-order interpolation of shape functions are suitable for addressing the problem of complex frequencies and modes caused by variable structural geometries.FindingsNumerical results are presented for moderately thick circular cylindrical shells with different geometrical factors (circumferential wave number, thickness-to-radius ratio, thickness-to-length ratio) to demonstrate the effectiveness, accuracy and reliability of the proposed method. The hp-version refinement uses fewer optimised meshes than h-version mesh refinement, and only one-step interpolation of the higher-order shape function yields the eigensolutions satisfying the accuracy requirement.Originality/valueThe proposed combination of methodologies provides a complete hp-version adaptive FEM for analysing the free vibration of moderately thick circular cylindrical shells. This algorithm can be extended to general eigenproblems and geometric forms of structures to solve for the frequency and mode quickly and efficiently.

Nonequilibrium molecular dynamics method based on coarse-graining formalism: Application to a nonuniform temperature field system.

A nonequilibrium molecular dynamics method is proposed to produce nonequilibrium states flexibly. In this method, virtual points are set in a simulation box, and coarse-grained physical quantities at these points are constrained using Gauss's principle of least constraint. The coarse-grained physical quantities are evaluated by averaging microscopic quantities with an appropriate weight. To obtain the weight to evaluate the coarse-grained physical quantities, a shape function matrix is initially constructed from the particle configuration. This matrix expresses an interpolation of the physical quantities at particle positions from the coarse-grained quantities at the virtual points. Then, a matrix form of the weight is calculated as the Moore-Penrose pseudoinverse matrix for the shape function matrix. This method is applied to constrain the coarse-grained kinetic energy and produce a nonuniform temperature field in the system. The temperature profile at a nonequilibrium steady state depends on the method for constructing the shape function matrix. In particular, a local temperature coincides with the coarse-grained temperature when the shape function matrix is constructed based on a higher-order interpolation.