A note on the convergence of finite element approximations for problems formulated in curvilinear coordinate systems

The asymptotic accuracies of structural shell theories are reviewed. Several finite element models to solve arch problems are formulated by utilizing the shell theories. The asymptotic rate of energy convergence is determined by the ability of the approximate strains to assume arbitray polynomial states. The optimal choice of interpolation functions for tangential and normal displacement is treated. Stress resultants and stress couples are evaluated by using nodal forces (strain integration method) or internal strain patterns (strain method). The accuracy of these methods are investigated. In particular the distribution of errors in the strains calculated by the strain method is determined and utilized for accuracte stress evaluation. The theoretical considerations are supported by a vaste number of numerical experiments, which confirm the theoretical results.     

Nodal finite element approximations for the neutron transport equation

Two classes of nodal methods, weakly and strongly discontinuous, are introduced and applied to the numerical solution of the neutron transport equation in two-dimensional Cartesian geometry and discrete ordinates. These methods are then applied for the approximation of the solution of a reference problem well known in the nuclear engineering literature.     

Reduced integration for improved accuracy of finite element approximations

For completeness the finite element bases which are used for approximate solutions of elliptic problems of order 2p by the Ritz method must include the functions corresponding to the constant value of the pth derivative. In actual usage, to ensure a positive definite system of algebraic equations, additional interpolating functions are introduced. This leads to “multiple covering” of some of the system modes and results in overestimation of stiffness. Reduced integration techniques eliminate some of this multiple covering and thereby give improved accuracy. Selective reduced integration has been found useful in the analysis of flexural problems. In this paper we suggest the use of only the minimal covering that is sufficient for convergence. A technique for solution of the discretized system is given. Numerical performance data show remarkable improvement over conventional procedures. The proposed scheme yields good approximation even for very coarse meshes. This indicates the possibility of considerable economy in the cost of obtaining finite element solutions to complex problems, e.g. coupled field problems, three-dimensional problems, stress concentration etc.     

A note on the finite element method for the space-fractional advection diffusion equation

In this paper, a note on the finite element method for the space-fractional advection diffusion equation with non-homogeneous initial-boundary condition is given, where the fractional derivative is in the sense of Caputo. The error estimate is derived, and the numerical results presented support the theoretical results.     

An analysis of natural convection using the thermal finite element discrete Boltzmann equation

The lattice Boltzmann method (LBM) is the simple numerical simulator for fluids because it consists of linear equations. Excluding the higher differential term, the LBM for a temperature field is also achieved as an easy numerical simulation method. However, the LBM is hardly applied to body fitted coordinates for its formulation. It is then difficult to calculate complex lattices using the LBM. In this paper, the finite element discrete Boltzmann equation (FEDBE) is introduced to deal with this weakness of the LBM. The finite element method is applied to the discrete Boltzmann equation (DBE) of the basic equation of the LBM. For FEDBE, the simulation using complex lattices is achieved, and it will be applicable for the development in engineering fields. The natural convection in a square cavity and the Rayleigh–Bernard convection are chosen as the test problem. Each simulation model is accurate enough for the flow patterns, the temperature distribution and the Nusselt number. This method is now considered good for the flow and temperature field, and is expected to be introduced for complex lattices using the DBE.     

On uniform convergence rates for Eulerian and Lagrangian finite element approximations of convection-dominated diffusion problems

Abstract Standard error estimates in the literature for finite element approximations of nonstationary convection-diffusion problems depend either reciprocally on the diffusion parameter or on higher order norms of the solution. Therefore, the estimates generally become worthless in the convection-dominated case with 0 < 1. In this work we develop a rigorous -uniform convergence theory for finite element discretizations of convection-dominated diffusion problems in Eulerian and Lagrangian coordinates. Here, the constants that arise in the error estimates depend on norms of the data and not of the solution and remain bounded in the hyperbolic limit 0. In particular in the Lagrangian case this requires modifications to standard finite element error analyses. In the Eulerian formulation -uniform convergence of order one half is proven whereas in the Lagrangian framework -uniform convergence of optimal order is established. The estimates are based on -uniform a priori estimates for the solution of the continuous problems which are derived first.     

A remark concerning divergence accuracy order for H(div)-conforming finite element flux approximations

The construction of finite element approximations in $\mathbf{H}(\text{div},\Omega )$ usually requires the Piola transformation to map vector polynomials from a master element to vector fields in the elements of a partition of the region $\Omega$. It is known that degradation may occur in convergence order if non affine geometric mappings are used. On this point, we revisit a general procedure for the improvement of two-dimensional flux approximations discussed in a recent paper of this journal (Comput. Math. Appl. 74 (2017) 3283–3295). The starting point is an approximation scheme, which is known to provide $L^2$-errors with accuracy of order $k+1$ for sufficiently smooth flux functions, and of order $r+1$ for flux divergence. An example is $R{T}_k$ spaces on quadrilateral meshes, where $r=k$ or $k?1$ if linear or bilinear geometric isomorphisms are applied. Furthermore, the original space is required to be expressed by a factorization in terms of edge and internal shape flux functions. The goal is to define a hierarchy of enriched flux approximations to reach arbitrary higher orders of divergence accuracy $r+n+1$ as desired, for any $n\ge 1$. The enriched versions are defined by adding higher degree internal shape functions of the original family of spaces at level $k+n$, while keeping the original border fluxes at level $k$. The case $n=1$ has been discussed in the mentioned publication for two particular examples. General stronger enrichment $n>1$ shall be analyzed and applied to Darcy’s flow simulations, the global condensed systems to be solved having same dimension and structure of the original scheme.     

Stability,convergence and accuracy of a new finite element method for the circular arch problem

The arch problem with shear deformation based upon the Hellinger-Reissner variational formulation is studied in a parameter-dependent form. A mixed Petrov-Galerkin method is used to construct a discrete approximation. Finite elements with equal-order discontinuous stress and continuous displacement interpolations, unstable in the Galerkin method, are proved to be stable in the new formulation. Error estimates indicate optimal rates of convergence for displacements and suboptimal rates, with gap one, for stresses. Numerical experiments confirm these estimates. The good accuracy of the mixed Petrov-Galerkin method is illustrated in some deep and shallow thin arch examples. No shear or membrane locking is present using full integration schemes.     

A viscoelastic-viscoplastic constitutive equation and its finite element implementation

A viscoelastic-viscoplastic constitutive model for isotropic materials undergoing isothermal infinitesimal deformation is proposed. The model is based on the assumption that the total strain rate is decomposable into a viscoelastic and a viscoplastic portion. Consequently, the model consists of a linear viscoelastic model in series with a modified plasticity model. This modified plasticity model adopts the classical Drucker-Prager yield surface with isotropic hardening and the associative flow rule of the invicid theory of plasticity. However, hardening is assumed to be a function of both the viscoplastic strain as well as the total strain rate. In this manner, the proposed model acquires the advantage of having both the initial and the subsequent yield surfaces to be a function of the strain rate, a property which has its experimental supportive evidence for viscoplastic materials such as polymers and some metals at highly elevated temperatures.A finite-element algorithm is developed to implement the constitutive equation derived in Part I. This algorithm adopts a combination of the tangent stiffness matrix and the initial load approach. The method of treating the transitional region between viscoelastic and viscoelastic-viscoplastic behavior is given. The details of implementation is described. Convergence of the computation scheme is discussed.Two examples are calculated numerically to demonstrate the strain rate and the pressure effects on the mechanical behavior of some viscoelastic-viscoplastic material. Results show that essential features in the stress-strain diagram obtained experimentally are exhibited by the model.     

A new mixed finite element based on different approximations of the minors of deformation tensors

Finite element formulations for arbitrary hyperelastic strain energy functions that are characterized by a locking-free behavior for incompressible materials, a good bending performance and accurate solutions for coarse meshes need still attention. Therefore, the main goal of this contribution is to provide an improved mixed finite element for quasi-incompressible finite elasticity. Based on the knowledge that the minors of the deformation gradient play a major role for the transformation of infinitesimal line-, area- and volume elements, as well as in the formulation of polyconvex strain energy functions a mixed finite element with different interpolation orders of the terms related to the minors is developed. Due to the formulation it is possible to condensate the mixed element formulation at element level to a pure displacement form. Examples show the performance and robustness of the element.     

A finite element numerical solution of natural convection in enclosed cavities

Steady state free convective flow enclosed within a cavity and subjected to a temperature gradient is predicted using the finite element method. The matrix equations resulting from the finite element discretisation and formulation are solved using both an iterative and a modified Newton-Raphson scheme. An assessment of the variation in the characteristics of the flow regime is made in association with the dimensionless Prandtl and Rayleigh numbers. A further parameter of interest in such problems is the cavity aspect ratio. The upper limit for the Rayleigh number (based on cavity width) presented in the present paper is 10⁷. The flow patterns are obtained for Prandtl numbers in the range 10^?2 ? Pr ? 10³ and for aspect ratios 1, 10, 20. Where possible the results are compared with existing solutions obtained using the finite difference method. A satisfactory correlation exists where such comparisons can be made. The results complement and extend those obtained during previous theoretical and numerical investigations.     

Maximum principle and uniform convergence for the finite element method

The solution of the Dirichlet boundary value problem over a polyhedral domain Ω ? Rⁿ, n ≥ 2, associated with a second-order elliptic operator, is approximated by the simplest finite element method, where the trial functions are piecewise linear. When the discrete problem satisfies a maximum principle, it is shown that the approximate solution u_h converges uniformly to the exact solution u if u ? W_1,p (Ω), with p > n, and that $\text{∥u?u}{}_{\mathrm{h}}\text{∥}{}_{\mathrm{L}}\text{∞(Ω) = O(h)}$ if u ? W²,p(Ω), with 2p > n. In the case of the model problem ?Δu+au = f in Ω, u = u_o on δΩ, with a ? 0, a simple geometrical condition is given which insures the validity of the maximum principle for the discrete problem.     

A wave equation model for finite element tidal computations

A shallow water wave equation is developed from the primitive two-dimensional shallow water equation. A finite element model based on this equation and the primitive momentum equation is developed. A finite difference formulation is used in the time domain which allows the model to be implicit or explicit while still centered in time. Results obtained with linear triangles and quadratic quadrilaterals are reported, and compare well with analytic solutions. The model incorporates all of the economical advantages of earlier models, and errors due to short wavelength spatial noise are suppressed without recourse to artificial means.     

A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation

To avoid the local oscillations that still remain using the streamline-upwind/Petrov-Galerkin formulation for the scalar convection-diffusion equation, the introduction of a nonlinear crosswind dissipation is proposed. It is shown that the method is less overdiffusive than other discontinuity-capturing techniques and has better numerical behavior. The design of the crosswind diffusion is based on the study of the discrete maximum principle for some simple cases.     

A posteriori error estimates for finite element discretizations of the heat equation

We consider discretizations of the heat equation by A-stable -schemes in time and conforming finite elements in space. For these discretizations we derive residual a posteriori error indicators. The indicators yield upper bounds on the error which are global in space and time and yield lower bounds that are global in space and local in time. The ratio between upper and lower bounds is uniformly bounded in time and does not depend on any step-size in space or time. Moreover, there is no restriction on the relation between the step-sizes in space and time.     

A Galerkin finite element method for time-fractional stochastic heat equation

In this study, a Galerkin finite element method is presented for time-fractional stochastic heat equation driven by multiplicative noise, which arises from the consideration of heat transport in porous media with thermal memory with random effects. The spatial and temporal regularity properties of mild solution to the given problem under certain sufficient conditions are obtained. Numerical techniques are developed by the standard Galerkin finite element method in spatial direction, and Gorenflo–Mainardi–Moretti–Paradisi scheme is applied in temporal direction. The convergence error estimates for both semi-discrete and fully discrete schemes are established. Finally, numerical example is provided to verify the theoretical results.     

A finite element formulation for elastoplasticity based on a three-field variational equation

A three-field variational equation, which expresses the momentum balance equation, the plastic consistency condition, and the dilatational constitutive equation in a weak form, is proposed as a basis for finite element computations in hardening elastoplasticity. The finite element formulation includes algorithms for the integration of the elastoplastic rate constitutive equations which are similar to members of the “return mapping” family of algorithms employed in displacement formulations, except that the proposed algorithms are not required to explicitly satisfy the plastic consistency condition at the end of each time step. This condition is imposed globally by the inclusion of a variational equation that suitably constrains the solution. The plastic incompressibility constraint is also treated in an appropriate variational sense. Solution of the nonlinear finite element equations is obtained by use of Newton's method and details of the linearization of the variational equation are given. The formulation is developed for an associative von Mises plasticity model with general nonlinear isotropic and kinematic strain hardening. A number of numerical test examples are provided.