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The crown pillar in the stope structure belongs to the category of thick plate, and its thickness determines the stability of the engineering structure. The key to determining the safe thickness of the crown pillar lies in solving its deflection function. Previous researchers often used the Galerkin method and other methods except the Ritz method to solve the deflection function of the crown pillar under simple boundary conditions. However, these methods are difficult to solve for the deflection function of the crown pillar under complex boundary conditions. Therefore, this paper builds upon the Reissner plate theory and introduces the Ritz method while considering the influence of strain components εz, γyz, and γzx on the flexural deformation of the crown pillar. Thus, the Reissner–Ritz method for solving the deflection function of the crown pillar in the stope is developed. Taking the failure of the crown pillar in stope 27 in the + 280 m section of the Daxin Manganese Mine as an example, firstly, the maximum tensile stress of the crown pillar is compared with its tensile strength to determine the safe thickness of the crown pillar in stope 27. The correctness of the chosen safe thickness for the crown pillar in stope 27 is verified using the Reissner–Ritz method. Then, FLAC3D is used to model and analyze the 8 m thick crown pillar in stope 27, and the accuracy of the Reissner–Ritz method in determining the safe thickness of the crown pillar is verified through finite element analysis. Finally, a comparison is made between the Reissner–Ritz method and the Galerkin method in solving the deflection function of the crown pillar under the uniformly distributed load, considering both simple and complex boundary conditions. The research results show that the Reissner–Ritz method has significant advantages over the Galerkin method in solving the deflection function of the crown pillar under the uniformly distributed load, even under complex boundary conditions. The findings are of great significance for solving the deflection function of thick plates under both simple and complex boundary conditions under the uniformly distributed load and determining the safe thickness of thick plates. Highlights: Based on the Reissner thick plate theory, the Ritz method is introduced, taking into account the influence of strain components εz, γyz, and γzx on the bending deformation of the crown pillar, forming the Reissner–Ritz method for solving the deflection function of the crown pillar in the stope. The range of safe thickness values for the crown pillar is determined based on the thickness-span ratio and safety factor, incorporating engineering experience to finalize the safe thickness of the crown pillar in the stope. The accuracy of the Reissner–Ritz method in determining the safe thickness of the four-sided fixed support crown pillar under the uniformly distributed load is verified through finite element analysis. Comparing and analyzing the calculation results of the Reissner Ritz method with the Galerkin method. [ABSTRACT FROM AUTHOR]

In this paper, we study a split-step Galerkin finite element (FE) method for the two-dimensional Riesz space-fractional coupled nonlinear Schrödinger equations (CNLSs). The proposed method adopts a second-order split-step technique to handle the nonlinearity and FE approximation to discretize the fractional derivatives in space, which avoids iteration at each time layer. The analysis of mass conservative and convergent properties for this split-step FE scheme is performed. To test its capability, some numerical tests and the simulation of the double solitons intersection and plane wave are carried out. The results and comparisons with the algorithm combined with Newton's iteration illustrate its effectiveness and advantages in computational efficiency. [ABSTRACT FROM AUTHOR]

This paper applies the spectral Galerkin method to numerically solve Riesz space-fractional convection–diffusion equations. Firstly, spectral Galerkin algorithms were developed for one-dimensional Riesz space-fractional convection–diffusion equations. The equations were solved by discretizing in space using the Galerkin–Legendre spectral approaches and in time using the Crank–Nicolson Leap-Frog (CNLF) scheme. In addition, the stability and convergence of semi-discrete and fully discrete schemes were analyzed. Secondly, we established a fully discrete form for the two-dimensional case with an additional complementary term on the left and then obtained the stability and convergence results for it. Finally, numerical simulations were performed, and the results demonstrate the effectiveness of our numerical methods. [ABSTRACT FROM AUTHOR]

We introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for the dual-porosity-Stokes problem. This coupled problem describes the interaction between free flow in macrofractures/conduits, governed by the Stokes equations, and flow in microfractures/matrix, governed by a dual-porosity model. We prove that the HDG method is strongly conservative, well-posed, and give an a priori error analysis showing dependence on the problem parameters. Our theoretical findings are corroborated by numerical examples. [ABSTRACT FROM AUTHOR]

We consider fully discrete finite element approximations for a semilinear optimal control system of partial differential equations in two cases: for distributed and Robin boundary control. The ecological predator-prey optimal control model is approximated by conforming finite element methods mimicking the spatial part, while a discontinuous Galerkin method is used for the time discretization. We investigate the sensitivity of the solution distance from the target function, in cases with smooth and rough initial data. We employ low, and higher-order polynomials in time and space whenever proper regularity is present. The approximation schemes considered are with and without control constraints, driving efficiently the system to desired states realized using non-linear gradient methods. [ABSTRACT FROM AUTHOR]

In the present work, we examine and analyze an hp-version interior penalty discontinuous Galerkin finite element method for the numerical approximation of a steady uid system on computational meshes consisting of polytopic elements on the boundary. This approach is based on the discontinuous Galerkin method, enriched by arbitrarily shaped elements techniques as has been introduced in [13]. In this framework, and employing extensions of trace, Markov-type, and H¹/L²-type inverse estimates to arbitrary element shapes, we examine a stationary Stokes uid system enabling the proof of the inf/sup condition and the hp- a priori error estimates, while we investigate the optimal convergence rates numerically. This approach recovers and integrates the exibility and superiority of the discontinuous Galerkin methods for uids whenever geometrical deformations are taking place by degenerating the edges, facets, of the polytopic elements only on the boundary, combined with the efficiency of the hp-version techniques based on arbitrarily shaped elements without requiring any mapping from a given reference frame. [ABSTRACT FROM AUTHOR]

• Vibration control of interconnected functionally graded fluid-conveying pipe is investigated. • A vibration experiment is established to explore NiTiNOL-steel damping ability. • Modal analysis is solved through the Rayleigh-Ritz approach and finite element method. • Investigation of amplitude-frequency response of the structure is carried out. • NiTiNOL-steel has significant nonlinear vibration control capabilities. In this study, an exploratory vibration experiment is established and an interconnected functionally graded material (FGM) pipeline system is installed. This experiment represents the first time that NiTiNOL-steel wire ropes (NiTi-ST) are applied to FGM pipeline system. And the experimental results demonstrate that the NiTi-ST has a practical vibration damping effect. Based on the experiment, a further theoretical pipeline system model is created. The effects of fluid velocity, temperature difference, and clamp stiffness on the modal and response of the pipeline system are added. The modal analysis part is solved through the implementation of the Rayleigh-Ritz approach using a set of modified orthogonal polynomials. In order to verify the accuracy of the Raylei-Ritz calculations, the finite element method (FEM) is also used. The response analysis part is solved through the Galerkin truncation method (GTM) and harmonic balance method (HBM). To match the harmonic balance solutions, the Runge-Kutta method (RK) are adopted. The results of the response analysis reveal that NiTi-ST demonstrates significant nonlinear vibration control capabilities to the interconnected FGM pipeline system under various conditions. [ABSTRACT FROM AUTHOR]

The accurate numerical solution at an acoustic–elastic interface is important for offshore exploration. The solution requires careful implementation for the acoustic–elastic boundary conditions. In this work, we leverage a nodal discontinuous Galerkin method, in which the unstructured uniform triangular meshes are used for the model meshing and an explicit upwind numerical flux derived from the Riemann problem is adopted to handle the boundary conditions at the acoustic–elastic interface. Several numerical results are provided to assess the accuracy and convergence properties of this method. The convergence analysis is carried out in the coupled model with a flat interface, and the accuracy of the proposed method is verified in the curved interface coupled model. Finally, a more complex model with a salt dome, inspired by real geophysical applications, is carried out in this study. The numerical results demonstrate that the proposed nodal discontinuous Galerkin method is effective and accurate for dealing with the coupled acoustic–elastic media with complex geometries. [ABSTRACT FROM AUTHOR]

We present in this article a numerical technique to simulate the hydrodynamic behaviour of a magnetic reading device. In the transient regime, the evolution of the air pressure is computed as the solution of a time dependent compressible Reynolds equation. The proposed numerical method is based on a local discontinuous Galerkin method combined with a $ \theta- $method for the time derivative. We present some numerical tests to show the good behaviour of the numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, we design and analyze an interior penalty hybridized discontinuous Galerkin (IP-HDG) method for the Darcy flow in the two- and three-dimensional fractured porous media. The discrete fracture model is used to model the fractures. The piecewise polynomials of degree |$k$| are employed to approximate the pressure in the fractures and the pressure in the surrounding porous media. We prove that the IP-HDG method is well posed if the penalty parameter is large enough. Based on the discrete solutions of pressures, the discrete Darcy velocity in the matrix and the reduced fractures can be recovered, respectively, to be locally mass-conservative. A robust residual-based a posteriori error estimator is established for an energy-norm of pressure. Finally, numerical results are provided to show the efficiency of the proposed a posteriori error estimator. [ABSTRACT FROM AUTHOR]

This article focuses on the nonlinear dynamics of transverse vibration of the axially moving yarn system. In this article, the motion of yarn in actual production is analyzed, and the transverse nonlinear vibration equation of the viscoelastic axially moving yarn system with Kelvin model is established by using Newton's second law. In the modeling process, the viscoelasticity of yarn material and the periodic fluctuation of tension in the process of movement are considered. The partial differential control equations of the system are transformed into ordinary differential equations by the Galerkin method. In the first three modes, when the speed is low, the related trial functions occupy the main part, and with the increase of speed, the trial functions of other orders occupy more proportion in the modal functions. Through numerical analysis, it is found that when the moving yarn is subjected to external excitation, the lateral vibration displacement has an increasing trend, and it is found that the nonlinear dynamic phenomenon alternates with the change of system parameters, such as single periodic motion, double periodic motion, and multiple periodic motion. Finally, an experimental platform for lateral vibration of yarn transmission was built and its working principle introduced. The lateral vibration of axially moving yarn was studied from the experimental point of view. It was proved that there are abundant phenomena of periodic motion, period doubling motion, and chaotic motion in the axially moving yarn system, which provides an experimental reference for theoretical research of lateral nonlinear vibration of axially moving yarn. [ABSTRACT FROM AUTHOR]

In this article, the spatial local discontinuous Galerkin (LDG) approximation coupled with the temporal implicit‐explicit Runge–Kutta (RK) evolution for the micropolar fluid equations are adopted to construct the discretization method. To avoid the incompressibility constraint, the artificial compressibility strategy method is used to convert the micropolar fluid equations into the Cauchy–Kovalevskaja type equations. Then the LDG method based on the modal expansion and the implicit‐explicit RK method are properly combined to construct the expected third‐order method. Theoretically, the unconditionally stable of the fully discrete method are derived in multidimensions for triangular meshs. And the numerical experiments are given to verify the theoretical and effectiveness of the presented methods. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical approximation to the one-dimensional Cahn-Hilliard equation with time periodic solution. We adopt the implicit Euler method and the spectral Galerkin method for the temporal and spatial discretization, respectively. Then the error estimates are proved for both the semi-discrete and the fully discrete schemes. Numerical experiments are carried out to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we study the semi-linear two-dimensional reaction–diffusion equation with Dirichlet boundaries. A reliable numerical scheme is designed, coupling the nonstandard finite difference method in the time together with the Galerkin in combination with the compactness method in the space variables. The aforementioned equation is analyzed to show that the weak or variational solution exists uniquely in specified space. The a priori estimate obtained from the existence of the weak or variational solution is used to show that the designed scheme is stable and converges optimally in specified norms. Furthermore, we show that the scheme preserves the qualitative properties of the exact solution. Numerical experiments are presented with a carefully chosen example to validate our proposed theory. [ABSTRACT FROM AUTHOR]

In this article, we present a new high‐order discontinuous Galerkin (DG) method, in which neither a penalty parameter nor a stabilization parameter is needed. We refer to this method as penalty‐free DG. In this method, the trial and test functions belong to the broken Sobolev space, in which the functions are in general discontinuous on the mesh skeleton and do not meet the Dirichlet boundary conditions. However, a subset can be distinguished in this space, where the functions are continuous and satisfy the Dirichlet boundary conditions, and this subset is called admissible. The trial solution is chosen to lie in an augmented admissible subset, in which a small violation of the continuity condition is permitted. This subset is constructed by applying special augmented constraints to the linear combination of finite element basis functions. In this approach, all the advantages of the DG method are retained without the necessity of using stability parameters or numerical fluxes. Several benchmark problems in two dimensions (Poisson equation, linear elasticity, hyperelasticity, and biharmonic equation) on polygonal (triangles, quadrilateral, and weakly convex polygons) meshes as well as a three‐dimensional Poisson problem on hexahedral meshes are considered. Numerical results are presented that affirm the sound accuracy and optimal convergence of the method in the L2$$ {L}^2 $$ norm and the energy seminorm. [ABSTRACT FROM AUTHOR]

The cantilevered micro-beam is an important component in the field of Atomic Force Microscope (AFM) and its behavior is governed by chaotic dynamics. To better understand these dynamics, a theoretical model based on the Euler–Bernoulli beam model has been developed and experiments have been conducted. The model incorporates both the modified couple stress theory (MCST) and external forces to account for the size-dependent effect and electrostatic load. The reduced model in one degree of freedom is obtained by the Galerkin method. The validity of the reduced model is tested by comparing the results from simulations and published results and experiments. The Melnikov method is then used to detect the chaotic threshold the cantilevered micro-beam. The numerical results test the effects of MCST on the complex responses of the micro-cantilever. Implications of the research can extend to the design and optimization of MEMS devices. [ABSTRACT FROM AUTHOR]

This paper deals with the theoretical studies to investigate the in-plane eigenvalue problems and the one-to-one-to-one internal resonance behaviors of a double-cable-stayed flexible beam-structure mounted with a manipulator system (DCFBMS) under constant rotational speed. A more precise and rigorous dynamic model with proper boundary conditions has been developed in which geometric nonlinearity and coupling nonlinearity of cables and beam are considered while the manipulator has been modeled as a rigid link. The in-plane eigenvalue problem of the model considering the boundary and continuous conditions has been evaluated and graphically demonstrated. Then, the ordinary differential equations (ODEs) are obtained by Galerkin's method and multiple scales method has been employed to further analyze the vibration attributes of steady-state responses and their stability under internal resonance condition. The nonlinear response, stability and bifurcations for one-to-one-to-one internal resonance have also been investigated by varying system parameters. Analytically obtained results have been verified by Runge–Kutta method and found to be in good agreement. The present theoretical results deliver a useful insight into the nonlinear dynamic behavior and operational stability of DCFBMS. [ABSTRACT FROM AUTHOR]

Principal-internal joint resonance of an axially moving beam with elastic constraints is investigated, where the beam is located in magnetic field excited by current-carrying wires. Based on the magnetoelasticity theory and Hamilton principle, the nonlinear magneto-elastic vibration equations are derived. The displacement function is obtained by the elastic constraint boundary condition, and then the equation is discretized into 2-DOF ordinary differential equations using the Galerkin integral method. The method of multiple scales is employed for obtaining the amplitude–frequency response equations with coupling of the first two vibration modes. Through examples, amplitudes varying with frequency tuning parameter, axial velocity, current intensity, and external excitation force are exhibited. Results indicate that as current intensity increases, electromagnetic damping increases and the amplitude decreases; As external excitation force increases and axial velocity decreases respectively, the amplitude increases and the system changes from single periodic motion to quasi-periodic motion and then to single periodic motion; The lower-order modes are always dominant when the principle-internal resonance occurs. [ABSTRACT FROM AUTHOR]

Numerical unsteady approach based on Galerkin method with the inclusion of adaptive mesh was executed in this paper to simulate the freezing of water within a container. The domain has two triangular and trapezoidal cold surfaces and two adiabatic walls. The governing equations show the domination of conduction and transient source term has been involved. Galerkin method for modeling was applied and good accommodation has been reported based on comparison with previous data. Adding nano-powders makes the amount of heat to be released with stronger conduction and freezing time decreases about 26.76% when m = 8. 6. Augmenting the fraction of powders can make the rate of process increase. For the greatest fraction of powders, an augmenting shape factor can enhance the speed of process about 6.95%. [ABSTRACT FROM AUTHOR]

Asphalt mixture is a viscoelastic material composed of aggregates, asphalt mastics, and voids. Meanwhile, the dynamic interaction between vehicle and bridge makes asphalt deck pavement in a more complex stress state, leading to premature damage such as cracks and ruts during its operation service period. Therefore, the effects of the viscoelastic characteristics of asphalt mixtures and vehicle‐bridge interactions on the dynamic behaviors of deck pavement cannot be ignored. However, it is difficult to directly couple the viscoelastic properties of mixtures and vehicle‐bridge interactions using the finite element method. Thus, this paper presents a decoupled method named the separation reconstruction algorithm. To theoretically verify whether the separate construction (SR) algorithm is consistent with the direct coupling (DC) algorithm, two dynamic system models of a Bernoulli–Euler composite beam under a moving vehicle are established, which correspond to the two algorithms. The kinematic equations of the system are derived and dispersed by the classical Galerkin method. The nonlinear integral term is simplified by the parity of the function, periodicity of the trigonometric function, and derivation. Then, the coupled equation can be rewritten in an ordinary differential form and solved with the Runge–Kutta method. Finally, it is proven that the proposed SR algorithm is basically consistent with the DC algorithm, and the SR algorithm is extended to practical engineering to study the coupling dynamic responses of viscoelastic deck pavement more accurately. Some new insight is gained by exploring the stress status of viscoelastic deck pavement under coupled conditions, which is conducive to more accurate evaluations of surface tensile effects induced by vehicle‐bridge interactions. [ABSTRACT FROM AUTHOR]

In this paper, a reliable Jacobi Galerkin method is developed and analyzed to solve a particular class of cordial Volterra integral equations. Existence and uniqueness theorems approve that these kinds of equations possess smooth solutions versus smooth data, so representing their Galerkin solutions based on suitable polynomial basis produces spectrally accurate approximations. Moreover, in order to control condition number growth, we designed a reliable approach that calculates the approximate solutions recursively without solving any high-conditioning systems. Indeed, this approach deals with solving equations in a long integration domain with even high oscillatory solutions. The convergence analysis of the presented approach is established, and the familiar spectral accuracy is justified in L ∞ -norm. Numerical results are presented to demonstrate the effectiveness of the proposed method. Finally, we provide an application of this method to approximate solution of a two-dimensional case. [ABSTRACT FROM AUTHOR]

In this article, the two-way coupled thermoelastic behavior of axially graded composite plate and doubly-curved (cylindrical, spherical, hyperbolic, and elliptical) panels is examined under conductive-convective boundary and uniformly distributed loading conditions. Here, the material properties of the axial functionally graded panel are computed by employing the power-law-based Voigt's scheme. The material and geometric nonlinearities are incorporated using the cubic-polynomial-based temperature-dependent constitutive model and higher-order kinematics-based Green-Lagrange strain, respectively. The temperature profile is obtained through thermally nonlinear theory associated with high temperature, which is used to extract the energy equations obtained from the first law of thermodynamics and deformation-dependent entropy relation. The weak forms of motion and heat-transfer equations are derived using Galerkin's method and further combined into the two-way coupled field equation through 2D-finite element approximation via Lagrangian elements. The transient responses are computed via the Newmark and the Crank-Nicolson schemes, whereas the nonlinear iterations are executed through Picard's iteration technique. The performance of the fully coupled model for an axially graded (metal/ceramic) structure is verified with analytical, numerical, and experimental results and tested through a variety of numerical illustrations under various sets of conditions. [ABSTRACT FROM AUTHOR]

The proposed work discusses discrete collocation and discrete Galerkin methods for second kind Fredholm–Hammerstein integral equations on half line [ 0 , ∞) using Kumar and Sloan technique. In addition, the finite section approximation method is applied to transform the domain of integration from [ 0 , ∞) to [ 0 , α ] , α > 0 . In contrast to previous studies in which the optimal order of convergence is achieved for projection methods, we attained superconvergence rates in uniform norm using piecewise polynomial basis function. Moreover, these superconvergence rates are further enhanced by using discrete multi-projection (collocation and Galerkin) methods. In order to support the provided theoretical framework, numerical examples are included as well. [ABSTRACT FROM AUTHOR]

This paper presents a nonstandard numerical manifold method (NMM) for solving Burgers' equation. Employing the characteristic Galerkin method, we initially apply the Crank–Nicolson method for temporal discretization along the characteristic. Subsequently, utilizing the Taylor expansion, we transform the semi-implicit formula into a fully explicit form. For spacial discretization, we construct the NMM dual-cover system tailored to Burgers' equation. We choose constant cover functions and first-order weight functions to enhance computational efficiency and exactly import boundary constraints. Finally, the integrated computing scheme is derived by using the standard Galerkin method, along with a Thomas algorithm-based solution procedure. The proposed method is verified through six benchmark numerical examples under various initial boundary conditions. Extensive comparisons with analytical solutions and results from alternative methods are conducted, demonstrating the accuracy and stability of our approach, particularly in solving Burgers' equation at high Reynolds numbers. [ABSTRACT FROM AUTHOR]

We consider the Galerkin and Collocation Isogeometric approximations of the acoustic wave equation with absorbing boundary conditions, while the time discretization is based on second-order Newmark schemes. A numerical study investigates the properties of the two IGA methods as concerns stability thresholds, convergence errors, accuracy, computational time, and sparsity of the stiffness matrices varying the polynomial degree p , mesh size h , regularity k , and time step Δ t. In order to compare the two IGA methods, we focus on two meaningful examples in the framework of wave propagation simulations: a test problem with an oscillatory exact solution having increasing wave number, and the propagation of two interfering Ricker wavelets. Numerical results show that the IGA Collocation method retains the convergence and stability properties of IGA Galerkin. In all examples considered, IGA Collocation is in general less accurate when we adopt the same choices of parameters p , k , h , and Δ t. On the other hand, regarding the computational cost and the amount of memory required to achieve a given accuracy, we observe that the IGA Collocation method often outperforms the IGA Galerkin method, especially in the case of maximal regularity k = p − 1 with increasing NURBS degree p. [ABSTRACT FROM AUTHOR]

Quadratic NURBS-based discretizations of the Galerkin method suffer from volumetric locking when applied to nearly-incompressible linear elasticity. Volumetric locking causes not only smaller displacements than expected, but also large-amplitude spurious oscillations of normal stresses. Continuous-assumed-strain (CAS) elements have been recently introduced to remove membrane locking in quadratic NURBS-based discretizations of linear plane curved Kirchhoff rods (Casquero et al. in Comput Methods Appl Mech Eng 360:112765, 2022). In this work, we propose two generalizations of CAS elements (named CAS1 and CAS2 elements) to overcome volumetric locking in quadratic NURBS-based discretizations of nearly-incompressible linear elasticity. CAS1 elements linearly interpolate the strains at the knots in each direction for the term in the variational form involving the first Lamé parameter while CAS2 elements linearly interpolate the dilatational strains at the knots in each direction. For both element types, a displacement vector with C 1 continuity across element boundaries results in assumed strains with C 0 continuity across element boundaries. In addition, the implementation of the two locking treatments proposed in this work does not require any additional global or element matrix operations such as matrix inversions or matrix multiplications. The locking treatments are applied at the element level and the nonzero pattern of the global stiffness matrix is preserved. The numerical examples solved in this work show that CAS1 and CAS2 elements, using either two or three Gauss–Legrendre quadrature points per direction, are effective locking treatments since they not only result in more accurate displacements for coarse meshes, but also remove the spurious oscillations of normal stresses. [ABSTRACT FROM AUTHOR]

Due to the considerable complexity of meshfree approximants, there still lacks a rational theoretical estimate on the frequency accuracy of lumped mass meshfree methods. In this study, a detailed theoretical analysis is presented for the lumped mass Galerkin meshfree formulation with particular reference to the frequency accuracy. The proposed accuracy analysis is quite general and is fully based upon the reproducing or consistency conditions of meshfree shape functions and their gradients. The theoretical findings reveal that there exists an important and interesting odd/even basis degree discrepancy feature for the frequency accuracy, i.e., (p + 2)th and (p + 1)th order accurate frequencies are attained by the lumped mass Galerkin meshfree formulation using even and odd degrees of basis functions, where p is the degree of basis function employed in the meshfree shape function construction. Furthermore, it is found that interpolatory meshfree shape functions are capable of not only circumventing the non-physical modes arising from the lumped mass Galerkin meshfree formulation using the standard non-interpolatory shape functions, but also elevating the frequency accuracy for meshfree structural vibration analysis. Numerical examples well replicate the theoretical frequency accuracy analysis results for lumped mass Galerkin meshfree methods. [ABSTRACT FROM AUTHOR]

In this work, we analyze a discontinuous Galerkin finite element method for the equations of motion that arise in the 2D Oldroyd model of order one. We investigate the existence and uniqueness of semidiscrete discontinuous solutions, as well as the consistency of the scheme. We derive new a priori and regularity results for the discrete solution and establish optimal error estimates in L∞$$ {L}^{\infty } $$‐norm in time and energy norm in space for the velocity and L2$$ {L}^2 $$‐norm in both time and space for the pressure. Uniform estimates are derived for sufficiently small data. We next apply the backward Euler method to the semidiscrete formulation and establish optimal fully discrete error estimates. At the end, we conduct numerical experiments to support our theoretical results and analyze the findings. [ABSTRACT FROM AUTHOR]

Nanotubes could be fabricated in different shapes for various applications, as they have the possibility of changing curvature. Vibration and stability of a curved nanotube conveying fluid and nanoparticle are investigated considering surface and non-local effects. The surface effects are considered in the nanotube inner and outer layers. The obtained equation is discretized using the Galerkin method. The main parameter effects on the stability and the first two modes are illustrated, including the total opening angle, the nanoparticle mass-ratio, the non-local parameter, the surface-effects parameter, and the nanoparticle position. They show significant effects on the dynamic behavior of the nanostructure. [ABSTRACT FROM AUTHOR]

In the presence of viscous dissipation and radiation effects, a numerical analysis is performed for an unsteady‐free convective, radiative, chemically reactive, radiation absorption, viscous, incompressible, and electrically conducting fluid passing through an exponentially accelerating vertical porous plate. Numerical methods are used to solve the collection of nondimensional governing equations and their boundary conditions. Graphs are used to study the effects of different physical characteristics on flow quantities. Tables are used to study the differences in skin friction, the Sherwood number, and the Nusselt number for the physical interest. The analysis and explanation of the fluctuation in radiation and flow parameters on velocity and temperature gradient profiles through graphs using the Galerkin technique is the novelty of this work. For the magnetic, Prandtl, heat sink, radiation, Schmidt, and chemical reaction parameters, skin friction is significantly higher; however, for the Grashof, modified Grashof, permeability, radiation absorption, Dufour, and Soret numbers, skin friction has the opposite tendency. [ABSTRACT FROM AUTHOR]

In this work, we derive a geometrically nonlinear plate model based on the Kirchhoff hypothesis and the large deflection hypothesis, and provide an error analysis of the corresponding element-free Galerkin method. The penalty method is employed to enforce boundary conditions. The error estimate explicitly depends on nodal spacing, number of monomial basis functions, continuity of weight functions, and penalty factors, which provides some practical choices among these key parameters in engineering applications. In addition, we offer guidance on selecting appropriate penalty factors to improve numerical accuracy. Numerical experiments, involving clamped square and circle plates with uniform and nonuniform nodes, as well as a plate model with a corner singularity, are presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we use the weighted discontinuous Galerkin finite element method to solve the dual-porosity-Brinkman coupling model. The dual-porosity model describes fluid flow in a dual-porosity medium composed of matrix and microfractures, while the Brinkman equation describes fluid flow in hydraulic fractures or conduits containing "debris filling" or "proppant filling". The two models are coupled by the four physically effective interface conditions, which are constructed based on the fundamental properties of the dual-porosity model and the Brinkman-Darcy model. We provide a weighted discontinuous discrete scheme for the model and derive the optimal error estimates. Through numerical experiments, we verify the properties of the model and the advantages of the numerical method, such as the optimal convergence rate of the numerical solution and the simulation of the flow characteristics around hydraulic fractures and conduits. [ABSTRACT FROM AUTHOR]

In view of the great importance of dynamical behavior prediction of nanostructures in contact with fluid and their vast range of applications in biomedical engineering, aerospace, etc., in this research, the free vibration of a Nanoscale Euler-Bernoulli rotating beam coupled with incompressible viscous fluid is studied. Small-scale effects are applied by using nonlocal elasticity theory. Using the Navier-Stokes relation, the interaction forces between the fluid and nanobeam are obtained. Governing differential equations have been solved by Galerkin method and the system vibrations frequency response has been obtained for clamped-free boundary condition. Based on the results of this research, nonlocal elasticity has a different effect on different vibration modes. The frequency of the nanobeam coupled with the fluid quickly increases when applying this theory, and the presence of fluid reduces the natural frequencies. [ABSTRACT FROM AUTHOR]

The dynamo actions of an electrically conducting fluid in a precessing sphere are investigated over a wide range of parameters by direct numerical simulation using a Galerkin spectral method. The focus of this work is to identify the most promising parameter regimes for the dynamo action and to investigate the characteristics of the magnetic field generated by precession. The influence of different nutation angles (30 ° , 60 ° , 90 °) and different precession ratios on the ability to drive dynamo action are investigated. The optimal angle for dynamo actions is found at 90 ° , followed by 60 ° with retrograde precession. A moderate precession ratio around 0.3 is shown to be more feasible for dynamo actions. A rich set of self-sustained dynamo solutions are obtained in the parameter space we explored, including steady, periodic, quasi-periodic, and turbulent dynamos. The structure of the generated magnetic fields is analyzed by using helical wave decomposition. None of the precession driven dynamos we obtained produce a predominantly dipolar field, contrary to the convection driven dynamos. The long-time evolution of the magnetic dipole moment is investigated and different types of polarity reversals are observed. [ABSTRACT FROM AUTHOR]

In this manuscript, we study a thin and narrow plate equation that models the deck of a suspension bridge that is subject to a Balakrishnan-Taylor damping and a strong damping. First, by using the Faedo Galerkin method, we prove the existence of both global weak and regular solutions. Second, we prove the exponential stability of the energy for regular solutions by combining the multiplier method and a well-known result of Komornik. [ABSTRACT FROM AUTHOR]

We present a hybridized discontinuous Galerkin (HDG) solver for general time‐dependent balance laws. In particular, we focus on a coupling of the solution process for unsteady problems with our anisotropic mesh refinement framework. The goal is to properly resolve all relevant unsteady features with the smallest possible number of mesh elements, and hence to reduce the computational cost of numerical simulations while maintaining its accuracy. A crucial step is then to transfer the numerical solution between two meshes, as the anisotropic mesh adaptation is producing highly skewed, non‐nested sequences of triangular grids. For this purpose, we adopt the Galerkin projection for the HDG solution transfer as it preserves the conservation of physically relevant quantities and does not compromise the accuracy of high‐order method. We present numerical experiments verifying these properties of the anisotropically adaptive HDG method. [ABSTRACT FROM AUTHOR]

This paper proposes a new numerical method for a fully-coupled, quasi-static thermo-poroelasticity model in a unified enriched Galerkin (EG) method framework. In our method, the mechanics sub-problem is solved using a locking-free EG method, and the flow and heat sub-problems are solved using a locally-conservative EG method. The proposed method offers mass and energy conservation properties with much lower costs than other methods with the same properties, including discontinuous Galerkin methods and mixed finite element methods. The well-posedness and optimal a priori error estimates are carefully derived. Several numerical tests confirm the theoretical optimal convergence rates and the mass and energy conservation properties of the new method. [ABSTRACT FROM AUTHOR]

Residual-type error estimators in energy error for the mortar staggered discontinuous Galerkin (DG) discretizations of second order elliptic equations were developed. The error estimator was proved to be reliable and efficient. An auxiliary function was defined, making it capable of decomposing the energy error into a conforming part and a nonconforming part, which can be combined with the well-known Scott-Zhang local quasi-interpolation operator, and the mortar discrete formulation yields an error estimator in energy error. Importantly, our analysis does not require any saturation assumptions, which are often needed in the literature. Several numerical experiments were presented to confirm our proposed theories. [ABSTRACT FROM AUTHOR]

This research aims to investigate the static aeroelastic characteristics of a slender straight 2D wing using aerodynamic strip theory. The finite element method is employed to determine the wing's divergence speed and aileron effectiveness, while Galerkin's method, based on the principle of virtual work is used to obtain the influence coefficient of the straight wing. The application of aerodynamic strip theory and finite span correction is utilized to establish a correlation between elastic twist and lift coefficient. Subsequently, a computational tool in MATLAB is formulated to derive an approximate solution for the static aeroelastic equilibrium equations concerning slender straight wings. An investigation is conducted into the impact of various elastic axis positions on the divergence speed and its implications for structural integrity are analyzed. It was observed in the study that the incorporation of finite span correction into the strip theory led to a 15% augmentation in the divergence speed of the slender wing. Validation of the mathematical model of the slender wing is performed through computational analyses conducted using ANSYS software. The flutter analysis examines parameters such as the distance between the elastic and aerodynamic axes, the sweep position, and the wing span. A MATLAB code is presented in the research article to explore the influence of these parameters on the flutter speed of a slender wing. Through an investigation of the interplay between these parameters and the flutter speed, the study strives to enhance comprehension of the fundamental mechanisms governing flutter occurrence in slender wings. The current research reveals that the flutter speed is notably affected by both the eccentricity and span of the wing. Specifically, a reduction in eccentricity leads to a 1.5% enhancement in flutter speed, while increasing the sweep angle from 15 to 30 degrees for a wing with a 15ft span results in a 2.54% increase in flutter speed. Moreover, wings spanning from 5ft to 15ft exhibit a 5% rise in flutter speed. These findings offer valuable insights for the design of more efficient and stable wings. [ABSTRACT FROM AUTHOR]

This article is concerned with the existence and uniqueness of results and the numerical modeling of a frictionless contact problem between a thermo-viscoelastic body and a rigid boundary condition. The contact is described by the normal compliance condition. We derive a weak formulation of the model and prove the existence of a unique weak solution using the Galerkin method and the Banach fixed point theorem. The fully discrete finite element scheme of the weak formulation is introduced, and error estimates for the approximate solution are derived. We present and study a successive iterative (decomposition) method to solve two sub-problems for the displacement field and the temperature sequentially. Finally, we present the results of the numerical simulations, demonstrating the method's performance. [ABSTRACT FROM AUTHOR]

The article presents the development of the weak Galerkin finite element method (WG-FEM) for semilinear hyperbolic problems. Semidiscrete error estimate in L 2 -norm as well as H 1 -norm have been executed for the weak Galerkin space (P k (K) , P k (∂ K) , [ P k - 1 (K) ] 2) , where k ≥ 1 is an integer. For a fully discrete scheme, we employ the Newmark scheme for temporal discretization. Finally, a few numerical results are provided to validate theoretical results. [ABSTRACT FROM AUTHOR]

[Display omitted] The results of shock-tube measurements of acetylene concentration and characteristics of the ensemble of soot particles during the pyrolysis of benzene, ethylene, ethylene–methane and ethylene–propane diluted with argon are compared with the predictions of the unified kinetic model of soot formation implemented in the MACRON program based on the Galerkin method. The simulation results demonstrate good agreement with experimental data for the pyrolysis of ethylene alone and with the specified additives, thereby confirming the two route mechanism of soot nuclei formation. The agreement for benzene pyrolysis is less satisfactory. [ABSTRACT FROM AUTHOR]

A Generalized Morse Potential (GMP) is an extension of the Morse Potential (MP) with an additional exponential term and an additional parameter that compensate for MP's erroneous behavior in the long range part of the interaction potential. Because of the additional term and parameter, the vibrational levels of the GMP cannot be solved analytically, unlike the case for the MP. We present several numerical approaches for solving the vibrational problem of the GMP based on Galerkin methods, namely, the Laguerre Polynomial Method (LPM), the Symmetrized LPM, and the Polynomial Expansion Method (PEM), and apply them to the vibrational levels of the homonuclear diatomic molecules B2, O2, and F2, for which high level theoretical near full configuration interaction (CI) electronic ground state potential energy surfaces and experimentally measured vibrational levels have been reported. Overall, the LPM produces vibrational states for the GMP that are converged to within spectroscopic accuracy of 0.01 cm−1 in between 1 and 2 orders of magnitude faster and with much fewer basis functions/grid points than the Colbert–Miller Discrete Variable Representation (CN-DVR) method for the three homonuclear diatomic molecules examined in this study. A Python library that fits and solves the GMP and similar potentials can be downloaded from https://gitlab.com/gds001uw/generalized-morse-solver. [ABSTRACT FROM AUTHOR]

Summary: This study shows how GPU performance of the ADER discontinuous Galerkin method in SeisSol (an earthquake simulation software) can be further improved while preserving its original design that ensures high CPU performance. We introduce a new code generator ("ChainForge") that fuses subsequent batched matrix multiplications ("GEMMs") into a single GPU kernel, holding intermediate results in shared memory as long as necessary. The generator operates as an external module linked against SeisSol's domain specific language YATeTo and, as a result, the original SeisSol source code remains mainly unchanged. In this paper, we discuss several challenges related to automatic fusion of GPU kernels and provide solutions to them. By and large, we gain ≈$$ \approx $$60% in performance of SeisSol's wave propagation solver using Fused‐GEMMs compared to the original GPU implementation. We demonstrated this on benchmarks as well as on a real production scenario simulating the Northridge 1994 earthquake. [ABSTRACT FROM AUTHOR]

This paper is concerned with a phase-field model driven by interface diffusion, which is an elliptic and nonlinear degenerate parabolic coupled system arising in a theory of interface motion in the elastically deformable materials. Spinodal decomposition is a typical example for this motion. We shall study the global existence of weak solutions to this model with a general polynomial potential function via the Galerkin method. Moreover, we perform the numerical simulations to investigate the spinodal decomposition process of binary alloy by applying this model. [ABSTRACT FROM AUTHOR]