Summary
Finite elements of class 𝒞1 are suitable for the computation of magnetohydrodynamics instabilities in tokamak plasmas. In addition, isoparametric approximations allow for a precise alignment ...of the mesh with the magnetic field line. Mesh alignment is crucial to achieve axisymmetric equilibria accurately. It is also helpful to deal with the anisotropy nature of magnetized plasma flows. In this numerical framework, several practical simulations are now available. They help to understand better the operation of existing devices and predict the optimal strategies for using the international ITER tokamak under construction. However, a mesh‐aligned isoparametric representation suffers from the presence of critical points of the magnetic field (magnetic axis, X‐point). We here explore a strategy that combines aligned mesh out of the critical points with non‐aligned unstructured mesh in a region containing these points. By this strategy, we can avoid highly stretched elements and the numerical difficulties that come with them. The mesh‐aligned interpolation uses bi‐cubic Hemite‐Bézier polynomials on a structured mesh of curved quadrangular elements. On the other hand, we assume reduced cubic Hsieh‐Clough‐Tocher finite elements on an unstructured triangular mesh. Both meshes overlap, and the resulting formulation is a coupled discrete problem solved iteratively by a suitable one‐level Schwarz algorithm. In this paper, we will focus on the Poisson problem on a two‐dimensional bounded regular domain. This elliptic equation is a simplified version of the axisymmetric tokamak equilibrium one at the asymptotic limit of infinite major radius (large aspect ratio).
Finite elements of class C1 are suitable for the computation of magnetohydrodynamics instabilities in tokamak plasmas. In addition, isoparametric approximations allow for a precise alignment of the mesh with the magnetic field line. However, a mesh‐aligned isoparametric representation suffers from the presence of critical points of the magnetic field (magnetic axis, X‐point). Here we explore a strategy that combines aligned mesh out of the critical points with non‐aligned unstructured mesh in a region containing these points. The mesh‐aligned interpolation uses bi‐cubic Hemite‐Bezier polynomials on a structured mesh of curved quadrangular elements, which allow to remain in the physical space. Reduced cubic Hsieh‐Clough‐Tocher finite elements are then adopted on an unstructured triangular mesh. Both meshes overlap, and the resulting formulation is a coupled discrete problem solved iteratively by a suitable one‐level Schwarz algorithm. The Poisson problem on a two‐dimensional bounded regular domain is considered as a simplified version of the axisymmetric tokamak equilibrium one at the asymptotic limit of infinite major radius (large aspect ratio). Numerical results on the accuracy of the local interpolation and of the coupled problem are provided together with an analysis of the algorithm convergence. This analysis is a first step to the adoption of the presented methodology in the industrial code such as JOREK.
This paper expands the isoparametric framework to construct a stable,
H
1
-conforming, and divergence-free method for the Stokes problem in two dimensions based on the Scott-Vogelius pair on ...Clough-Tocher splits. The pressure space is defined through composition, whereas the velocity space is constructed via a new divergence-preserving mapping that imposes full continuity across shared edges in the isoparametric mesh. Our construction is motivated by operators and spaces found in isoparametric
C
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finite element methods. We prove the method is stable, pressure-robust, and has optimal order convergence. Numerical experiments are provided which confirm the theoretical results.
The influence of the internal debonding on the structural stiffness and the nonlinear modal characteristics of the layered structure are examined extensively in the current research article. For the ...investigation purpose, the shell frequency responses are obtained numerically for both the linear and the nonlinear cases via a generic type of mathematical formulation using the Equivalent Single Layer (ESL) theory in the framework of two kinematic models. The current formulation not only includes the influence of the transverse shear deformations but also satisfies the parabolic variation of transverse shear stress through the thickness. Additionally, the geometrical nonlinear distortion modeled via Green–Lagrange strain–displacement relations. Further, the internal debonding between the adjacent layers are modeled using sub-laminate approach and the displacement continuity between segments (laminate and delaminate) have been established through the intermittent continuity conditions. The nonlinear system governing equation of the vibrated structure is obtained via Hamilton’s principle and converted to set of nonlinear algebraic equations through the isoparametric finite element (FE) steps. The desired responses are solved numerically with the help of robust (direct iterative method) technique and compared with available results to demonstrate the solution accuracy. Subsequently, an adequate number of examples are solved for the delaminated structure using the current higher-order nonlinear models and the influential parameters discussed in detail.
This paper presents the nonlinear free vibration behavior and response of functionally graded plates and shell panels under various boundary conditions. The displacement formulation followed uses a ...sinusoidal non-polynomial quasi-3D higher order shear deformation theory with six variables and, the parabolic transverse stress distribution functions are used to model stresses in the thickness direction. Present formulation employs the Green-Lagrange nonlinear strains along with Sander's approximation to account for larger flexural response which could be experienced by the structure. An eight noded C
0
continuous isoparametric finite element is used for numerical implementation of nonlinear finite element solver developed in MATLAB. Developed numerical formulation is employed to study and analyze the nonlinear free vibration response of the spherical, cylindrical and hyperboloid shell panels and the obtained results are seen to be consistent and good agreement with the literature thus establishing the effectiveness and accuracy of the present work.
•Solution to cross-anisotropic viscoelastic saturated soils is derived based on the Merchant model.•Solution for a slowly changing load is taken as the kernel function for the BEM of the soils.•The ...Mindlin plate model is established by 8-node isoparametric finite element method.•Interaction solution of the problem is proposed using the BEM-FEM coupling and substructure technique.•Influence parameters on the time-dependent behavior of the raft are investigated.
By introducing the elastic-viscoelastic correspondence principle and the integral transform technique, the extended precise integral solution to Biot's consolidation equation is derived for cross-anisotropic viscoelastic saturated soils based on the Merchant model. A displacement-time solution for a slowly changing load is obtained based on the above solution in the Laplace transformed domain, which is taken as the kernel function for the boundary element method (BEM). The Mindlin plate model is established by 8-node isoparametric finite element method (FEM), and the substructure condensation technique is employed to couple the stiffness matrices of the superstructure and the plate. With the BEM-FEM coupling method, a semi-analytical and semi-numerical solution is proposed for the interaction between layered cross-anisotropic viscoelastic saturated soils and raft foundation considering the stiffness contributions of the superstructure. Numerical examples are performed to study the influences of viscoelastic parameters, cross-anisotropic parameters, raft thickness and superstructure stiffness on the time-dependent behavior of the settlement and bending moment for the raft.
The thermal buckling load factors of damaged (crack) laminated composite structures have been investigated numerically first-hand. Their strength reversal due to the functional material reinforcement ...under the elevated environmental conditions is reported in detail. In this regard, a generic finite element model of the damaged layered structure has been derived in the higher-order kinematic model considering the Green–Lagrange type of strain (to count the large deformation due to temperature). Further, the shape memory alloy (SMA) fibre effect has been introduced in the proposed mathematical model via marching technique under the change in temperature to achieve the modified elastic property. The final governing equation of buckled structure has been derived and solved through isoparametric finite element steps. The final buckling temperatures of the damaged layered structural system with and without SMA (volume fraction and prestrain) fibres were obtained, and the results indicate a good improvement (maximum up to 37%). Additionally, the structural geometry-related parameters, including their shapes, have been changed to show the model's applicability.
Abstract
This paper provides a unified error analysis for nonconforming space discretizations of linear wave equations in the time domain. We propose a framework that studies wave equations as ...first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite-dimensional Hilbert spaces. A lift operator maps the semidiscrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and nonconforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the acoustic wave equation.
In the present work, nonlinear free vibration analysis of composite laminated plates and shallow cylindrical/spherical/hyperboloid shell panels considering geometric nonlinearity is carried out using ...non-polynomial inverse trigonometric higher-order shear deformation theory with seven degrees of freedo (DOFs). The present theory assumes parabolic variation of out-of-plane stresses and satisfies traction-free boundary conditions on the top and bottom surfaces of the composite as a priori. A nonlinear finite element model is developed and applied to obtain discretized nonlinear equations. The geometric nonlinearity in sense of Green-Lagrange considering von-Kármán assumptions is incorporated in formulation. An eight noded efficient C
0
continuous isoparametric rectangular finite element is implemented in the present nonlinear analysis. The efficacy and accuracy of the present theory and finite element model is validated with the available literature results. For the analysis, various types of plates and shell panels with different material properties, lamination schemes, thickness ratios, aspect ratios, modular ratios, curvature ratios, and boundary conditions are considered. The effects of these different geometric and material properties on nonlinear frequency ratios at various amplitude ratios are examined in detail.
In 3D braided composites, some fibers are oriented in thickness directions, increasing strength and reducing de-lamination ability. The equivalent elastic properties of the 3D braided plate are ...computed using a volume averaging approach. Here, the low-velocity impact of the 3D braided plate is investigated at various locations based on the 3D shear deformations theory incorporated with twelve-degree freedom per node using an eight nodded isoparametric finite element method. In this theory, transverse displacement is the function of the thickness coordinates, which is the unique advantage of this theory. The various parametric studies are performed by varying geometrical and material parameters.
•New staggered space–time DG schemes for the incompressible Navier–Stokes equations.•Arbitrary high order of accuracy for incompressible fluids in both space and time.•Staggered formulation allows to ...avoid Riemann solvers in several terms.•New space–time pressure-correction algorithm, leading to very sparse pressure system.•Linear system solvable with a matrix-free GMRES algorithm without preconditioner.
In this paper we propose a novel arbitrary high order accurate semi-implicit space–time discontinuous Galerkin method for the solution of the two dimensional incompressible Navier–Stokes equations on staggered unstructured triangular meshes. Isoparametric finite elements are used to take into account curved domain boundaries. The discrete pressure is defined on the primal triangular grid and the discrete velocity field is defined on an edge-based staggered dual grid. While staggered meshes are state of the art in classical finite difference approximations of the incompressible Navier–Stokes equations, their use in the context of high order DG schemes is novel and still quite rare. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse four-point block system for the scalar pressure, which is conveniently solved with a matrix-free GMRES algorithm. Note that the same space–time DG scheme on a collocated grid would lead to ten non-zero blocks per element, since substituting the discrete velocity into the discrete continuity equation on a collocated mesh would involve also neighbors of neighbors. From numerical experiments we find that our linear system is well-behaved and that the GMRES method converges quickly even without the use of any preconditioner, which is a unique feature in the context of high order implicit DG schemes. A very simple and efficient Picard iteration is then used in order to derive a space–time pressure correction algorithm that achieves also high order of accuracy in time, which is in general a non-trivial task in the context of high order discretizations for the incompressible Navier–Stokes equations. The flexibility and accuracy of high order space–time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both, space and time. The use of a staggered grid allows to avoid the use of Riemann solvers in several terms of the discrete equations and significantly reduces the total stencil size of the linear system that needs to be solved for the pressure. The proposed method is verified for approximation polynomials of degree up to p=4 in space and time by solving a series of typical numerical test problems and by comparing the obtained numerical results with available exact analytical solutions or other numerical reference data.
To the knowledge of the authors, this is the first time that a space–time discontinuous Galerkin finite element method is presented for the incompressible Navier–Stokes equations on staggered unstructured grids.
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