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The interest in phononic crystals and acoustic metamaterials has been an intensive subject of research in recent years. Finding a robust way to significantly expand or actively control the bandgap has received extensive attention. In this study, we propose a prestressed metamaterial beam attached with multiply local resonators connected by actively tunable piezoelectric springs. The Euler–Bernoulli beam theory and Timoshenko beam theory are applied in the theoretical analysis of the system. Further, the spectral element method is utilized to analytically compute the dispersion relation and transmission ratio and excellent agreement with reference to the benchmark is reported. The influences of an external axial force on the bandgap range and attenuation behavior are further studied. Subsequently, the effect of resonator number and mass on the local resonance bandgap structure is investigated in two parametric studies. The active control of bandgap range and frequency is then verified. By analyzing frequency response function, the tunable transmission ratio of a supercell can be observed. To conclude, this paper not only provides a guideline for designs of wave attenuation with multiple frequency regimes in a one-dimensional system, but it can also be extended to sub-wavelength wave manipulation designs. [ABSTRACT FROM AUTHOR]

This paper deals with form-finding and free vibration analysis of a pre-stressed class-one triplex tensegrity structure. The form-finding is performed via a two-step procedure, the nodal coordinates connectivity matrix, and structural element force density determination. Accordingly, the possible states for the nodal coordinates and the structural force density of the triplex prism have been determined by trial and error (based on topology and member type knowledge) to satisfy the force density, and equilibrium matrices rank requirements. Based on different structural topologies, the equation of the motion in the frequency domain for free vibration analysis of the system is derived using the spectral element approach and dynamic shape functions. Simulations are provided for different system heights and the top-bottom aria ratios and compared with the FEM. The numerical simulations in the form of a comparative study of the natural frequencies of triplex tensegrity prism with different heights and cross-sections represent the system’s robustness with different topologies for single or multi-stage applications.

The inverse problem for radiative transfer is important in many applications, such as optical tomography and remote sensing. Major challenges include large memory requirements and computational expense, which arise from high-dimensionality and the need for iterations in solving the inverse problem. Here, to alleviate these issues, we propose adaptive-mesh inversion: a goal-oriented hp -adaptive mesh refinement method for solving inverse radiative transfer problems. One novel aspect here is that the two optimizations (one for inversion, and one for mesh adaptivity) are treated simultaneously and blended together. By exploiting the connection between duality-based mesh adaptivity and adjoint-based inversion techniques, we propose a goal-oriented error estimator, which is cheap to compute, and can efficiently guide the mesh-refinement to numerically solve the inverse problem. We use discontinuous Galerkin spectral element methods to discretize the forward and the adjoint problems. Then, based on the goal-oriented error estimator, we propose an hp -adaptive algorithm to refine the meshes. Numerical experiments are presented at the end and show convergence speed-up and reduced memory occupation by the goal-oriented mesh adaptive method. [ABSTRACT FROM AUTHOR]

In this article we present the nonconforming spectral element method (NSEM) for linear second-order ordinary differential equations with boundary conditions having analytic solutions, and the interface problem. A fully discrete spectral element method for one-dimensional parabolic problems with smooth solutions and parabolic interface problem also have been considered. The stability and error estimates of NSEM for a two-point boundary value problem with analytic solutions are derived. The numerical formulation is described and it is essentially a least-squares formulation and spectral elements are nonconforming. The normal equations which arise in the minimization of the functional are solved using preconditioned conjugate gradient method without storing the stiffness matrix and load vector. The method is exponentially accurate. In the fully discrete formulation for the parabolic problems, Crank–Nicolson scheme is used in time variable and higher-order spectral elements are used in spatial variable. This method is second-order accurate in time and exponential accurate in spatial variable. Various numerical examples are presented to show the accuracy of the method. Finally, we review the nonconforming spectral element method for various problems in higher dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the approximation of abstract thermoelastic models. It is by now well known that approximated systems are not in general uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal in this paper is to study the uniform exponential/polynomial stability of a sequence of a system of weakly coupled thermoelastic models. We prove that when 0 ≤ β < 1 2 , the total energy of solutions is not uniformly exponentially stable, but it decays uniformly polynomially to zero. Finally, the results are applied to space semi-discretizations of thermoelastic beam equation in a bounded interval with homogeneous Dirichlet boundary conditions. We consider finite element, spectral element and finite difference semi-discretizations. Finally, we illustrate the mathematical results with several numerical experiments. [ABSTRACT FROM AUTHOR]

This paper proposes a new shell element formulation using the scaled boundary finite element (SBFE) method. A shell element is treated as a three-dimensional continuum. Its bottom surface is approximated with a quadrilateral spectral element and the shell geometry is represented through normal scaling of the bottom surface. Neumann expansion is applied to approximate the inversions of the matrix polynomials of the thickness coordinate ξ, including the Jacobian matrix and the coefficient of the second-order term in the SBFE equation. This permits the solution along the thickness to be expressed as a matrix exponential function whose exponent is a high-order matrix polynomial of ξ. After introducing the boundary conditions on the top and bottom surfaces and evaluating the resulting matrix exponential via Padé expansion, we derive the element stiffness and mass matrices. Poisson thickness locking is avoided fundamentally. Numerical examples demonstrate the applicability and efficiency of the formulation. [ABSTRACT FROM AUTHOR]

We consider a model coupling the Darcy equations in a porous medium with the Navier-Stokes equations in the cracks, for which the coupling is provided by the pressure's continuity on the interface. We discretize the coupled problem by the spectral element method combined with a nonoverlapping domain decomposition method. We prove the existence of solution for the discrete problem and establish an error estimation. We conclude with some numerical tests confirming the results of our analysis. [ABSTRACT FROM AUTHOR]

The non-destructive testing (NDT) plays a crucial role in geotechnical engineering and geophysical applications, especially in the design of earthquake-resistant foundations, geotechnical field investigation, and material characterization and detection of underground anomaly. Currently, the existing signal interpretation methods in NDT measurements still predominantly rely on empirical relations or subjective judgements. In this paper, we present the GeoNDT software, which is developed to provide an advanced physics-based signal interpretation method for NDT characterization of multiphase geomaterials. GeoNDT is able to model the propagation of stress waves and dispersion relations in dry (elastodynamic), saturated (two-phase poroelastodynamic), and three-phase frozen (multiphase poroelastodynamic) geomaterials using the meshless spectral element method. GeoNDT is flexible, general-purpose, and can be used seamlessly for advanced signal interpretation in geophysical laboratory testing including the bender element and ultrasonic pulse velocity tests, characterization of complex multiphase geomaterials, and in situ shallow seismic geophysics including the falling weight deflectometer and multichannel analysis of surface waves tests. The advanced physics-based signal interpretation feature of GeoNDT allows the quantitative characterization of geophysical and geomechanical properties of geomaterials and multilayered geosystems independently without making any simplified assumptions as common in the current practice. [ABSTRACT FROM AUTHOR]

The topological optimization technique is an effective way to provide the optimal design for the polymeric stents. Conventional topology optimization procedure for polymeric stent was all based on classical theory of elasticity and usually only used one layer of shell element in structural analysis. It will lead to that the simulated stent structural stiffness deviates from the real case. Thus, a novel 3D topology optimization procedure based on Cosserat theory is proposed to maximize the radial stiffness of polymeric stent by taking advantage of the size effect of polymer materials. The stent structural analysis is implemented by a proposed high-order Cosserat spectral element method, ensuring that the analysis and optimization results can achieve high accuracy with only one layer of element for the polymeric stent. Cosserat characteristic length of polymeric material is obtained by the three-point bending test, and a parametric analysis is conducted to identify the effect of Cosserat characteristic length, volume fraction, and unit cell size on the optimized stent structure, and the optimal results based on the Cosserat theory are compared with the conventional ones. It is found that the size effect has a significant impact on the topology optimization results of polymeric stent. Compared with conventional commercial stent, the optimal polymeric stent obtained by the Cosserat theory has lower volume fraction and higher radial stiffness. Our results provide an effective way to further improve the mechanical performance of the polymeric stent. [ABSTRACT FROM AUTHOR]

Efficient structural element-based models are essential for fast simulations of wave propagation in composite structures for model-based and physics-informed data-driven structural health monitoring. This article introduces the first efficient multiphysics time-domain spectral structural element for wave propagation analysis of beam and panel-type composite structures with piezoelectric transducers (patch or full layers) containing delaminations. A general framework is presented to model multiple delaminations and transducer patches located arbitrarily. The intact host and patch transducer-bonded laminates and sub-laminates between delaminations are modelled separately using an electromechanically coupled efficient layerwise zigzag theory (ZIGT) for kinematics and a piecewise quadratic variation for the electric potential across piezoelectric layers. The high-order spectral element (SE) features a virtual electric node to model equipotential surfaces of piezoelectric transducers apart from the usual physical nodes having mechanical and internal electric degrees of freedom. A hybrid point-least squares continuity approach is employed to maintain continuity at the intersections of delaminated sub-laminates or the patch-bonded laminate with the host laminate. The model's performance in capturing electroelastic waves' interaction with delamination is examined with reference to the conventional finite element (FE) solution based on the ZIGT, continuum-based FE solutions, and the SE solution based on a non-layerwise version of the laminate theory. Finally, the model is used to examine the impact of interfacial location and size of delaminations on wave propagation behaviour. • First efficient multiphysics spectral structural element for smart composite strips with delaminations. • Enforces nonlinear displacement field continuity at delamination and piezo patch intersections. • Employs virtual electric node to satisfy equipotential conditions on piezo surfaces. • Yields faster computation and better accuracy compared to the standard FE counterpart. • It will benefit guided wave-based structural health monitoring of delaminated structures. [ABSTRACT FROM AUTHOR]

A nonconforming spectral element method for the Stokes problem on nonsmooth domains has been proposed in Mohapatra et al. (J Comput Appl Math 372:112696, 2020). The main focus of this article is to study the performance of this method for Stokes problems on smooth curvilinear domains and Stokes problem with mixed boundary conditions. Various test cases are considered including the generalized Stokes problem in R 2 and R 3 to verify the exponential accuracy of the method. [ABSTRACT FROM AUTHOR]

In this work, we study the material and geometric uncertainty effects on the static, free vibration and dynamic behaviour of sandwich beam structures. A higher-order sandwich panel theory is considered for the analysis. The elastic properties of the sandwich beam are considered as 1-D non-Gaussian random field which causes local variation in mass and stiffness matrices of the beam. The discretization of the non-Gaussian random fields is performed using the expansion optimal linear estimation. To perform the numerical analysis, Monte-Carlo simulation along with the computationally efficient time-domain spectral element method is proposed. Numerical results are obtained for different boundary conditions as well as for different materials in the sandwich face sheet and core. Results obtained in this work quantify the effects of material and geometric uncertainty in the response behaviour of a sandwich beam. The individual effect of core thickness and Poisson's ratio of the core on the static, free vibration and dynamic response is quantified. It is observed that the uncertainties in material and geometric properties along with loading and boundary conditions influence the static, free vibration and dynamic response. [ABSTRACT FROM AUTHOR]

Multiphase flow can be difficult to simulate with high accuracy due to the wide range of scales associated with various multiphase phenomena. These scales may range from the size of individual particles to the entire domain of interest. Traditionally, large scale systems can only be simulated using averaging approaches that filter out the locations of individual particles. In this work, the Euler–Lagrange method is used to simulate large-scale dense particle systems in which each individual particle is tracked. In order to accomplish this, the highly scalable spectral element code nek5000 has been extended to handle the multiple levels of multiphase coupling in these systems. These levels include what has been called one-, two-, and four-way coupling. Here, each level has been separated to detail the computational impact of each stage. A binned ghost particle algorithm has also been developed to efficiently handle the challenges of two- and four-way coupling in a parallel processing context. The algorithms and their implementations are then shown to scale to 65,536 Message Passing Interface (MPI) ranks in both the strong and weak limits. After this, validation is performed through simulation of a small-scale fluidized bed. Lastly, a large-scale fluidized bed is simulated with 65,536 MPI ranks and is able to capture the unique physics of the onset of fluidization. [ABSTRACT FROM AUTHOR]

The approach of the next-generation computing platforms offers a tremendous opportunity to advance the state-of-the-art in global atmospheric dynamical models. We detail our incremental approach to utilize this emerging technology by enhancing concurrency within the High-Order Method Modeling Environment (HOMME) atmospheric dynamical model developed at the National Center for Atmospheric Research (NCAR). The study focused on improvements to the performance of HOMME which is a Fortran 90 code with a hybrid (MPIOpenMP) programming model. The article describes the changes made to the use of message passing interface (MPI) and OpenMP as well as single-core optimizations to achieve significant improvements in concurrency and overall code performance. For our optimization studies, we utilize the "Cori" system with an Intel Xeon Phi Knights Landing processor deployed at the National Energy Research Supercomputing Center and the "'Cheyenne" system with an Intel Xeon Broadwell processor installed at the NCAR. The results from the studies, using "workhorse" configurations performed at NCAR, show that these changes have a transformative impact on the computational performance of HOMME. Our improvements have shown that we can effectively increase potential concurrency by efficiently threading the vertical dimension. Further, we have seen a factor of two overall improvement in the computational performance of the code resulting from the single-core optimizations. Most notably from the work is that our incremental approach allows for high-impact changes without disrupting existing scientific productivity in the HOMME community. [ABSTRACT FROM AUTHOR]

The climate of stratosphere was simulated with a spectral element model under two different vertical level distributions, one had 28 levels with 0.3 hPa at the top of the atmosphere and another 66 levels with 4.5×10-6 hPa at the top boundary. Simulation results were validated against ERA-Interim reanalysis dataset from ECMWF (European Center for Medium-Range Weather Forecasts) and the influence of vertical level distributions on the numerical simulation of stratospheric temporal mean state had been analyzed. It is shown that the lift of upper boundary can reduce the temperature error above the height of 100 hPa level significantly. It can also improve the simulation of seasonal variation of zonal mean zonal wind on 50 hPa in the northern high latitudes. In both configurations, cold biases of 10-year averaged zonal mean of temperature are mainly discerned at levels around 75 hPa at low latitudes and levels around 175 hPa at high southern latitudes. The simulated seasonal variation of westerly at 50 hPa in the northern hemisphere with 66-layer model coincides with results of ERA-Interim better. [ABSTRACT FROM AUTHOR]

The under integration of the volume terms in the discontinuous Galerkin spectral element approximation introduces errors at non-conforming element faces that do not cancel and lead to free-stream preservation errors. We derive volume and face conditions on the geometry under which a constant state is preserved. From those, we catalog eight constraints on the geometry that preserve a constant state. Numerical examples are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

The application of the composite materials in the aeronautical and aerospace industries has been increasing over the last several decades. Compared to conventional metallic materials, they present better strength to weight and stiffness to weight ratio. However, they can also present a high level of uncertainty, mainly associated with the manufacturing processes. Besides the uncertainty in the composite material parameters, which can play a role in the structural dynamic response, randomness can also be associated with boundary condition and external excitation sources. This paper treats the dynamic analysis of a composite beam under random excitation and uncertainties in the boundary condition. The beam is modelled by the spectral element method, a wave propagation technique. Some numerical examples are used to study the influence of random source on the dynamic behaviour of the composite structure. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in constructing a scheme solving compressible Navier–Stokes equations, with desired properties including high order spatial accuracy, conservation, and positivity-preserving of density and internal energy under a standard hyperbolic type CFL constraint on the time step size, e.g., Δ t = O (Δ x). Strang splitting is used to approximate convection and diffusion operators separately. For the convection part, i.e., the compressible Euler equation, the high order accurate postivity-preserving Runge–Kutta discontinuous Galerkin method can be used. For the diffusion part, the equation of internal energy instead of the total energy is considered, and a first order semi-implicit time discretization is used for the ease of achieving positivity. A suitable interior penalty discontinuous Galerkin method for the stress tensor can ensure the conservation of momentum and total energy for any high order polynomial basis. In particular, positivity can be proven with Δ t = O (Δ x) if the Laplacian operator of internal energy is approximated by the Q k spectral element method with k = 1 , 2 , 3. So the full scheme with Q k (k = 1 , 2 , 3) basis is conservative and positivity-preserving with Δ t = O (Δ x) , which is robust for demanding problems such as solutions with low density and low pressure induced by high-speed shock diffraction. Even though the full scheme is only first order accurate in time, numerical tests indicate that higher order polynomial basis produces much better numerical solutions, e.g., better resolution for capturing the roll-ups during shock reflection. • The hyperbolic subproblem is solved by a positivity-preserving RKDG method. • The parabolic subproblem is solved by classical schemes to achieve positivity. • A proper choice of the IPDG method achieves total conservation. • Numerically a spatially higher order scheme produces better solutions. [ABSTRACT FROM AUTHOR]