Low-order Finite Element Methods Research Articles

Low-order finite element method for the static and vibration analysis of an antenna pedestal

In this work, an effective algorithm is developed for the static and vibration analysis of antenna pedestal, which can well balance the modelling efficiency and computational accuracy. At first, the antenna pedestal is discretized into a set of low-order tetrahedral elements that can be automatically generated. Then, the face-based smoothing domains are constructed based on the faces of the low-order tetrahedral mesh. With the help of strain smoothing technique, the system stiffness matrices are calculated in these smoothing domains and finally, the static and vibration responses of the antenna pedestal are obtained based on the related system equations. There is no need to add any penalty parameters or additional degrees of freedom. Numerical results demonstrate that the present algorithm can achieve much better accuracy and higher convergence rate than traditional FEM, which can be regarded as an efficient tool for the structural analysis of antenna pedestal.

A Spectral/hp-Based Stabilized Solver with Emphasis on the Euler Equations

The solution of compressible flow equations is of interest with many aerospace engineering applications. Past literature has focused primarily on the solution of Computational Fluid Dynamics (CFD) problems with low-order finite element and finite volume methods. High-order methods are more the norm nowadays, in both a finite element and a finite volume setting. In this paper, inviscid compressible flow of an ideal gas is solved with high-order spectral/hp stabilized formulations using uniform high-order spectral element methods. The Euler equations are solved with high-order spectral element methods. Traditional definitions of stabilization parameters used in conjunction with traditional low-order bilinear Lagrange-based polynomials provide diffused results when applied to the high-order context. Thus, a revision of the definitions of the stabilization parameters was needed in a high-order spectral/hp framework. We introduce revised stabilization parameters, τsupg, with low-order finite element solutions. We also reexamine two standard definitions of the shock-capturing parameter, δ: the first is described with entropy variables, and the other is the YZβ parameter. We focus on applications with the above introduced stabilization parameters and analyze an array of problems in the high-speed flow regime. We demonstrate spectral convergence for the Kovasznay flow problem in both L1 and L2 norms. We numerically validate the revised definitions of the stabilization parameter with Sod’s shock and the oblique shock problems and compare the solutions with the exact solutions available in the literature. The high-order formulation is further extended to solve shock reflection and two-dimensional explosion problems. Following, we solve flow past a two-dimensional step at a Mach number of 3.0 and numerically validate the shock standoff distance with results obtained from NASA Overflow 2.2 code. Compressible flow computations with high-order spectral methods are found to perform satisfactorily for this supersonic inflow problem configuration. We extend the formulation to solve the implosion problem. Furthermore, we test the stabilization parameters on a complex flow configuration of AS-202 capsule analyzing the flight envelope. The proposed stabilization parameters have shown robustness, providing excellent results for both simple and complex geometries.

Analysis of a Stabilised Finite Element Method for Power-Law Fluids

A low-order finite element method is constructed and analysed for an incompressible non-Newtonian flow problem with power-law rheology. The method is based on a continuous piecewise linear approximation of the velocity field and piecewise constant approximation of the pressure. Stabilisation, in the form of pressure jumps, is added to the formulation to compensate for the failure of the inf-sup condition, and using an appropriate lifting of the pressure jumps a divergence-free approximation to the velocity field is built and included in the discretisation of the convection term. This construction allows us to prove the convergence of the resulting finite element method for the entire range r>frac{2 d}{d+2} of the power-law index r for which weak solutions to the model are known to exist in d space dimensions, d in {2,3}.

A hybrid spectral/finite element method for accurate and efficient modelling of crack-induced contact acoustic nonlinearity

We propose a 2D hybrid spectral/finite element scheme for numerically resolving crack-induced contact acoustic nonlinearity in solid structures. While the high-order spectral element method (SEM) is much more efficient than the classical low-order finite element method (FEM), the fact that spectral elements are relatively large renders the SEM ineffective in discretizing microscopic cracks. The classical FEM, on the other hand, is good at modelling complex geometries. However, the computation of large-scale structures by the classical FEM can be extremely expensive. In this work, the coupling between high-order spectral elements and low-order finite elements is formulated by the Lagrange multipliers method. The nonlinear contact between crack surfaces is modelled using a penalty method. The aim of the proposed hybrid method is to simultaneously reduce the degrees of freedom (DOFs) in systems and ensure high numerical accuracy. A comprehensive mesh convergence study has been conducted to demonstrate the high convergence rate of the proposed method in modelling ultrasonic wave propagation, and its capability to converge for weak crack-induced contact acoustic nonlinearity. Through a series of numerical experiments on generation and propagation of nonlinear ultrasonic wave modes, the proposed method has been shown to possess a high accuracy and geometric flexibility, and require significantly less computational resources than the classical FEM. The proposed hybrid method will provide an efficient, accurate and robust numerical approach for studying crack-induced contact acoustic nonlinearity which nowadays is being widely exploited by non-destructive testing applications.

A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics

We present the application of a class of deep learning, known as Physics Informed Neural Networks (PINN), to inversion and surrogate modeling in solid mechanics. We explain how to incorporate the momentum balance and constitutive relations into PINN, and explore in detail the application to linear elasticity, and illustrate its extension to nonlinear problems through an example that showcases von Mises elastoplasticity. While common PINN algorithms are based on training one deep neural network (DNN), we propose a multi-network model that results in more accurate representation of the field variables. To validate the model, we test the framework on synthetic data generated from analytical and numerical reference solutions. We study convergence of the PINN model, and show that Isogeometric Analysis (IGA) results in superior accuracy and convergence characteristics compared with classic low-order Finite Element Method (FEM). We also show the applicability of the framework for transfer learning, and find vastly accelerated convergence during network re-training. Finally, we find that honoring the physics leads to improved robustness: when trained only on a few parameters, we find that the PINN model can accurately predict the solution for a wide range of parameters new to the network—thus pointing to an important application of this framework to sensitivity analysis and surrogate modeling.

Extended framework of Hamilton’s principle for single-degree-of-freedom nonlinear damping systems

The paper explores application of the variational formalism called extended framework of Hamilton’s principle to nonlinear damping systems. Single-degree-of-freedom systems with dominant source of nonlinearity from polynomial powers of the velocity are initially considered. Appropriate variational formulation is provided, and then the corresponding weak form is discretized to produce a novel computational method. The resulting low-order temporal finite element method utilizes non-iterative algorithm, and some examples are provided to verify its performance. The present temporal finite element method using small time step is equivalent to the adaptive Runge–Kutta–Fehlberg method with default error tolerances in MATLAB, and additional simulation shows its good convergence characteristics.

A Stabilized Low Order Finite Element Method for Three Dimensional Elasticity Problems

A Stabilized Low Order Finite Element Method for Three Dimensional Elasticity Problems

Partition of Unity Finite Element Analysis of Nonlinear Transient Diffusion Problems Using p-Version Refinement

We propose a high-order enriched partition of unity finite element method for linear and nonlinear time-dependent diffusion problems. The solution of this class of problems often exhibits non-smooth features such as steep gradients and boundary layers which can be very challenging to recover using the conventional low-order finite element methods. A class of steady-state exponential functions has been widely used for enrichment and its performance to numerically solve these challenges has been demonstrated. However, these enrichment functions have been used only in context of the standard h-version refinement or the so-called q-version refinement. In this paper we demonstrate that the p-version refinement can also be a very attractive option in terms of the efficiency and the accuracy in the enriched partition of unity finite element method. First, the transient diffusion problem is integrated in time using a semi-implicit scheme and the semi-discrete problem is then integrated in space using the p-version enriched finite elements. Numerical results are presented for three test examples of timedependent diffusion problems in both homogeneous and heterogeneous media. The computed results show the significant improvement when using the p-version refined enriched approximations in the finite element analysis. In addition, these results support our expectations for a robust and high-order accurate enriched partition of unity finite element method.

Analysis of Shear Wall Systems Using Plane Stress Elements

Finite element method takes advantage of robust methodology or analysis of different types of structures, particularly the tall buildings. Accordingly, a number of displacement and strain-based finite element approaches have been developed in various fields of civil engineering. However, many lower-order finite element methods still encounter analytical drawbacks dealing with analysis of tall buildings. The absence of an appropriate in-plane rotational stiffness in some of the lower-order finite element analysis methods and the existence of parasitic shear effects in the governing displacement functions are counted as the most effective factors which are likely to emerge within analysis of tall buildings. In this respect, finite element approach is presented in this paper, which is capable of incorporating the stiffness of all lateral load-resisting systems. Furthermore, as the horizontal strains are ignored in all elevation levels of the element, a uniform lateral displacement pattern is obtained for the proposed panel element. This panel-type element is a strain-based element including eight degrees of freedom which has been formulated according to the general behaviour of beam element. Consequently, in order to validate the accuracy and efficiency of the proposed approach, a number of numerical analyses are performed.

Weak imposition of constraints for structural membranes in transient geometrically nonlinear isogeometric analysis on multipatch surfaces

Membranes have been extensively used for the design of architectural and general structural models due to their low cost and high load carrying capacity. Traditionally such models were discretized using the standard low order Finite Element Method (FEM) which typically results in a compromised description of the geometry. However, the accurate geometric description of membrane structures is essential as for instance bifurcation points in geometrically nonlinear analysis may be inaccurately predicted when the geometric description of the model is not accurate enough. Moreover, the design of membrane structures typically requires several cycles of form-finding and subsequent structural analysis under various loads which can benefit from a direct connection to the Computer-Aided Design (CAD) environment using its exact geometric description. In this contribution, the form-finding analysis using the Updated Reference Strategy (URS) and the geometrically nonlinear transient analysis of membranes is extended to Isogeometric Analysis (IGA) on multipatch surfaces with Non-Uniform Rational B-Splines (NURBS). As typical in IGA for real CAD geometries, multiple patches with non-matching parametrizations are considered and therefore the continuity of the solution field along with the application of weak Dirichlet boundary conditions need to be addressed. Thus, four different constraint enforcement methods are elaborated and compared, namely, the Penalty, the Lagrange Multipliers, the augmented Lagrange Multipliers and a Nitsche-type method. For the latter method, a solution dependent stabilization approach is employed in order to render the Nitsche-type method coercive. All methods are elaborated and systematically compared in both form-finding analysis, whenever necessary, and subsequently in geometrically nonlinear transient analysis. It should be noted that the Nitsche-type method is more computationally demanding amongst these methods due to the additional nonlinear terms. However, the results suggest that the Nitsche-type method is advantageous for these kinds of problems as no parameter or discretization other than the isogeometric discretization within each patch needs to be specified prior to the analysis.

A Least-Squares/Relaxation Method for the Numerical Solution of the Three-Dimensional Elliptic Monge\u2013Amp\xe8re Equation

In this article, we address the numerical solution of the Dirichlet problem for the three-dimensional elliptic Monge–Ampère equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities. Dedicated numerical solvers are derived for the efficient solution of the local optimization problems with cubicly nonlinear equality constraints. The approximation relies on mixed low order finite element methods with regularization techniques. The results of numerical experiments show the convergence of our relaxation method to a convex classical solution if such a solution exists; otherwise they show convergence to a generalized solution in a least-squares sense. These results show also the robustness of our methodology and its ability at handling curved boundaries and non-convex domains.

A low-order finite element method for three dimensional linear elasticity problems with general meshes

The paper is concerned with a low-order finite element method, namely the staggered cell-centered finite element method, which has been proposed and analyzed in Ong et al. (2015) for two-dimensional compressible and nearly incompressible linear elasticity problems. In this work, we extend the results to the three-dimensional case and focus on the creating of the meshes. In particular, from a general primal mesh M, we construct a polygonal dual mesh M∗ and its submesh M∗∗ in a way such that each dual control volume of M∗ corresponds to a primal vertex and is a union (macro-element) of some fixed number of adjacent tetrahedral elements of M∗∗. The displacement is approximated by piecewise trilinear functions on the subdual mesh M∗∗ and the pressure by piecewise constant functions on the dual mesh M∗. As for two-dimensional case, such construction of the meshes and approximation spaces satisfies the macroelement condition, which implies stability and convergence of the scheme. Numerical experiments are carried out to investigate the performance of the proposed method on various mesh types.

QTT-finite-element approximation for multiscale problems I: model problems in one dimension

Tensor-compressed numerical solution of elliptic multiscale-diffusion and high frequency scattering problems is considered. For either problem class, solutions exhibit multiple length scales governed by the corresponding scale parameter: the scale of oscillations of the diffusion coefficient or smallest wavelength, respectively. As is well-known, this imposes a scale-resolution requirement on the number of degrees of freedom required to accurately represent the solutions in standard finite-element (FE) discretizations. Low-order FE methods are by now generally perceived unsuitable for high-frequency coefficients in diffusion problems and high wavenumbers in scattering problems. Accordingly, special techniques have been proposed instead (such as numerical homogenization, heterogeneous multiscale method, oversampling, etc.) which require, in some form, a-priori information on the microstructure of the solution. We analyze the approximation properties of tensor-formatted, conforming first-order FE methods for scale resolution in multiscale problems without a-priori information. The FE methods are based on the dynamic extraction of principal components from stiffness matrices, load and solution vectors by the quantized tensor train (QTT) decomposition. For prototypical model problems, we prove that this approach, by means of the QTT reparametrization of the FE space, allows to identify effective degrees of freedom to replace the degrees of freedom of a uniform "virtual" (i.e. never directly accessed) mesh, whose number may be prohibitively large to realize computationally. Precisely, solutions of model elliptic homogenization and high-frequency acoustic scattering problems are proved to admit QTT-structured approximations whose number of effective degrees of freedom required to reach a prescribed approximation error scales polylogarithmically with respect to the reciprocal of the target Sobolev-norm accuracy ź with only a mild dependence on the scale parameter. No a-priori information on the nature of the problems and intrinsic length scales of the solution is required in the numerical realization of the presently proposed QTT-structured approach. Although only univariate model multiscale problems are analyzed in the present paper, QTT structured algorithms are applicable also in several variables. Detailed numerical experiments confirm the theoretical bounds. As a corollary of our analysis, we prove that for the mentioned model problems, the Kolmogorov n-widths of solution sets are exponentially small for analytic data, independently of the problems' scale parameters. That implies, in particular, the exponential convergence of reduced basis techniques which is scale-robust, i.e., independent of the scale parameter in the problem.

A High-Order Finite Element Method for the Linearised Euler Equations

Sound propagation in complex non-uniform mean flows is an important feature of turbofan exhaust noise radiation. The Linearised Euler Equations are able to represent the strong shear layer refraction effects on the sound field, as well as multiple length scales. Frequency domain solvers are suitable for tonal noise and considered a way to avoid linear instabilities, which may occur with time domain solvers. However, the classical Finite Element Method suffers from dispersion error and high memory requirements. These shortcomings are particularly critical for high frequencies and for the Linearised Euler Equations, which involve up to five unknowns. In this paper, a high-order Finite Element Method is used to solve the Linearised Euler Equations in the frequency domain in order to overcome those issues. The model involves high-order polynomial shape functions, unstructured triangular meshes, numerical stabilisation and Perfectly Matched Layers. The acoustic radiation from a straight circular semi-infinite hard-wall duct with several mean flow configurations is computed. Comparisons with analytic solutions demonstrate the method accuracy. The acoustic and vorticity waves are well represented, as well as the refraction of the sound field across the jet shear layer. The high-order approach allows to use coarse meshes, while maintaining a sufficient accuracy. The benefits in terms of memory requirements are significant when compared to standard low-order Finite Element Method.

Finite Element Analysis of Viscoelastic Nanofluid Flow with Energy Dissipation and Internal Heat Source/Sink Effects

A numerical study is conducted of laminar viscoelastic nanofluid polymeric boundary layer stretching sheet flow. Viscous dissipation, surface transpiration (suction/injection), internal heat generation/absorption and work done due to deformation are incorporated using a second grade viscoelastic non-Newtonian nanofluid with non-isothermal associated boundary conditions. The nonlinear boundary value problem is solved using a higher order finite element method. The influence of viscoelasticity parameter, Brownian motion parameter, thermophoresis parameter, Eckert number, Lewis number, Prandtl number, internal heat generation and also wall suction on thermofluid characteristics is evaluated in detail. Validation with earlier non-dissipative studies is also included. The hp-finite element method achieves the desired accuracy at $$p=8$$ with comparatively less CPU cost per iteration (with less degrees of freedom, DOF) as compared to lower order finite element methods. The simulations have shown that greater polymer fluid viscoelasticity $$({k}_{1})$$ accelerates the flow. A rise in Brownian motion parameter (Nb) and thermophoresis parameter (Nt) elevates temperatures and reduce the heat transfer rates (local Nusselt number function). Increasing Eckert number increases temperatures whereas increasing Prandtl number (Pr) strongly lowers temperatures. Increasing internal heat generation $$({Q} > 0)$$ elevates temperatures and reduces the heat transfer rate (local Nusselt number function) whereas heat absorption $$({Q} < 0)$$ generates the converse effect. Increasing suction $$({f}_{w}>0)$$ reduces velocities and temperatures but elevates enhances mass transfer rates (local Sherwood number function), whereas increasing injection $$({f}_{w}<0)$$ accelerate the flow, increases temperatures and depresses wall mass transfer rates. The study finds applications in rheological nano-bio-polymer manufacturing.

Investigation of the effects of viscous damping mechanisms on structural characteristics in coupled shear walls

This study addresses energy dissipation mechanisms to investigate the effects of the internal and external viscous damping on structural characteristics in coupled shear walls. A discrete Reference Beam (RB) is firstly proposed and a Distributed Internal Viscous Damping (DIVD), composed by bending and shear mechanisms, is defined. Meanwhile, the linear classical damping is considered. A low-order finite element method (FEM) is adopted for lateral analyses. For the sake of simplicity, a Generalized Sandwich Beam (GSB) is then developed through the replacement of the set of connecting beams of the RB by an equivalent elastic and dissipative core and a FEM is employed for its dynamic analysis. The passive damping modeling through the GSB is presented using continuous models. Concerning slender structures, the GSB is reduced to a Coupled Two-Beam (CTB) including the damping effects. The analytical solution of the CTB is presented to be a benchmark for the FE solutions. The validity of both numerical and analytical solutions is confirmed via numerical examples. The effectiveness of proposed damping models on dynamical responses and vibration characteristics are tested with respect to continuum-based controlling parameters, and a qualitative model is consequently proposed to appropriately choose the damping mechanisms depending upon the parameters of coupled shear walls. The suitability of various damping models is finally compared to current damping predictors and full-scale measured data given for RC buildings. The results reveal that the bending and shear damping are somehow efficient where the linear classical damping is incapable to be always a proper mechanism.

Finite Element Analysis of a Bending Moment Formulation for the Vibration Problem of a Non-homogeneous Timoshenko Beam

In this paper we analyze a low-order finite element method for approximating the vibration frequencies and modes of a non-homogeneous Timoshenko beam. We consider a formulation in which the bending moment is introduced as an additional unknown. Optimal order error estimates are proved for displacements, rotations, shear stress and bending moment of the vibration modes, as well as a double order of convergence for the vibration frequencies. These estimates are independent of the beam thickness, which leads to the conclusion that the method is locking free. For its implementation, displacements and rotations can be eliminated leading to a well posed generalized matrix eigenvalue problem for which the computer cost of its solution is similar to that of other classical formulations. We report numerical experiments which allow us to assess the performance of the method.

A stabilized mixed finite element method based on g-biorthogonal systems for nearly incompressible elasticity

We present a finite element method for nearly incompressible elasticity using a mixed formulation of linear elasticity in the displacement–pressure form. The idea of stabilization of an equal order interpolation for Stokes equations is combined with biorthogonality to get rid of the bubble functions. A Petrov–Galerkin formulation for the pressure equation is used, where the trial and test spaces are different and form a g-biorthogonal system. This novel approach leads to a displacement-based low order finite element method for nearly incompressible elasticity for simplicial, quadrilateral and hexahedral meshes. Numerical results are provided to demonstrate the efficiency of the approach.

Low-order finite element method for the well-posed bidimensional Stokes problem

We work on a low-order finite element approximation of the vorticity, velocity and pressure formulation of the bidimensional Stokes problem. In a previous paper, we have introduced the adequate space in which to look for the vorticity in order to have a well-posed problem. In this paper, we deal with the numerical approximation of this space, prove optimal convergence of the scheme and show numerical experiments in good accordance with the theory. We remark that despite one supplementary unknown(the vorticity), results of the scheme are much better than the ones obtained with the P1 plus bubble−P1 element in the velocity–pressure formulation.

A very fast and accurate boundary element method for options with moving barrier and time-dependent rebate

A numerical method to price options with moving barrier and time-dependent rebate is proposed. In particular, using the so-called Boundary Element Method, an integral representation of the barrier option price is derived in which one of the integrand functions is not given explicitly but must be obtained solving a Volterra integral equation of the first kind. This equation is affected by several kinds of singularities, some of which are removed using a suitable change of variables. Then the transformed equation is solved using a low-order finite element method based on product integration. Numerical experiments are carried out showing that the proposed method is extraordinarily fast and accurate. In particular a high level of accuracy is achieved also when the initial price of the underlying asset is close to the barrier, when the barrier and the rebate are not differentiable functions, or when the optionʼs maturity is particularly long.