In Acosta etal. (2017), a complete n-dimensional finite element analysis of the homogeneous Dirichlet problem associated to a fractional Laplacian was presented. Here we provide a comprehensive and ...simple 2D MATLAB® finite element code for such a problem. The code is accompanied with a basic discussion of the theory relevant in the context. The main program is written in about 80 lines and can be easily modified to deal with other kernels as well as with time dependent problems. The present work fills a gap by providing an input for a large number of mathematicians and scientists interested in numerical approximations of solutions of a large variety of problems involving nonlocal phenomena in two-dimensional space.
Hybrid‐Trefftz stress elements for plate bending Teixeira de Freitas, João António; Tiago, Carlos
International journal for numerical methods in engineering,
15 May 2020, Volume:
121, Issue:
9
Journal Article
Peer reviewed
Summary
The hybrid‐Trefftz stress element is used to emulate conventional finite elements for analysis of Kirchhoff and Mindlin‐Reissner plate bending problems. The element is hybrid because it is ...based on the independent approximation of the stress‐resultant and boundary displacement fields. The Trefftz variant is consequent on the use of the formal solutions of the governing Lagrange equation to approximate the stress‐resultant field. In order to emulate conventional elements, nodal functions are used to approximate the displacements on the boundary of the element. Duality is used to set up the element solving system. The associated variational statements and conditions for existence and uniqueness of solutions are recovered. Triangular and quadrilateral elements are tested and characterized in terms of convergence, sensitivity to shear‐locking, and shape distortion. Their relative performance is assessed using assumed strain Mixed Interpolation of Tensorial Components (MITC) elements and recently proposed Trefftz‐based elements. This relative assessment is extended to a hypersingular problem to illustrate the effect of enriching the domain and boundary approximation bases.
A continuous space‐time Galerkin method is newly proposed for the numerical solution of inverse dynamics problems. The proposed space‐time finite element method is combined with servo‐constraints to ...partially prescribe the motion of the underlying mechanical system. The new approach to the feedforward control of infinite‐dimensional mechanical systems is motivated by the classical method of characteristics. In particular, it is shown that the simultaneous space‐time discretization is much better suited to solve the inverse dynamics problem than the semi‐discretization approach commonly applied in structural dynamics. Representative numerical examples dealing with elastic strings undergoing large deformations demonstrate the capabilities of the newly devised space‐time finite element method.
In this work, we propose a novel formulation for the solution of partial differential equations using finite element methods on unfitted meshes. The proposed formulation relies on the discrete ...extension operator proposed in the aggregated finite element method. This formulation is robust with respect to the location of the boundary/interface within the cell. One can prove enhanced stability results, not only on the physical domain, but on the whole active mesh. However, the stability constants grow exponentially with the polynomial order being used, since the underlying extension operators are defined via extrapolation. To address this issue, we introduce a new variant of aggregated finite elements, in which the extension in the physical domain is an interpolation for polynomials of order higher than two. As a result, the stability constants only grow at a polynomial rate with the order of approximation. We demonstrate that this approach enables robust high-order approximations with the aggregated finite element method. The proposed method is consistent, optimally convergent, and with a condition number that scales optimally for high order approximation.
In this work, we analyse the links between ghost penalty stabilisation and aggregation-based discrete extension operators for the numerical approximation of elliptic partial differential equations on ...unfitted meshes. We explore the behaviour of ghost penalty methods in the limit as the penalty parameter goes to infinity, which returns a strong version of these methods. We observe that these methods suffer locking in that limit. On the contrary, aggregated finite element spaces are locking-free because they can be expressed as an extension operator from well-posed to ill-posed degrees of freedom. Next, we propose novel ghost penalty methods that penalise the distance between the solution and its aggregation-based discrete extension. These methods are locking-free and converge to aggregated finite element methods in the infinite penalty parameter limit. We include an exhaustive set of numerical experiments in which we compare weak (ghost penalty) and strong (aggregated finite elements) schemes in terms of error quantities, condition numbers and sensitivity with respect to penalty coefficients on different geometries, intersection locations and mesh topologies.
•Links between strong (aggregated) and weak (ghost penalty) methods for C0 Lagrangian unfitted finite elements.•Discussion about the locking phenomenon of ghost penalty strategies in the strong limit.•Design and analysis of novel locking-free ghost penalty methods based on discrete extension.•Detailed numerical comparison of weak ghost penalty and strong aggregation-based schemes.•Superior accuracy and reduced sensitivity to parameters of novel approaches.
•At present corner point singularities are only partly understood.•Results from highly accurate finite element analyses have not resolved the problem.•Two dimensional analyses of sheets and plates ...are inherently approximations.•Attempts to find a three dimensional stress function for corner points have failed.•In the absence of a theoretical breakthrough the situation is unlikely to improve.
The linear elastic analysis of homogeneous, isotropic cracked bodies started in the 1900s. The existence of three dimensional corner point effects in the vicinity of a corner point where a crack front intersects a free surface was investigated in the late 1970s. An approximate solution by Bažant and Estenssoro explained some features of corner point effects but there were various paradoxes and inconsistencies. Results derived from finite element models showed that the analysis is incomplete. The stress field in the vicinity of a corner point appears to be the sum of two singularities.
When dealing with innovative materials – such as composites and metamaterials with complex microstructure – or structural components with non-orthogonal beam/plate geometry, the Finite Element Method ...can become very costly in calculations and time because of the use of very fine 3D meshes. By exploiting the Node-Dependent Kinematic approach of the Carrera Unified Formulation and using Lagrange expanding functions, this work presents the implementation of non-conventional 1D and 2D elements mainly based on the 3D integration of the approximating functions and computation of 3D Jacobian matrix inside the element for the derivation of stiffness and mass matrices; substantially, the resulting elements are 3D elements in which the order of expansion can be different in the three spatial directions. The free vibration analysis of some typical components is performed and the results are provided in terms of natural frequencies. The present elements allow us to accurately study beam-like and plate-like structures with non-orthogonal geometries by employing much less degrees of freedom with respect to the use of classical 3D finite elements.
•Non-conventional 1D and 2D elements based on the computation of 3D Jacobian matrix.•Present elements exploit the Node-Dependent Kinematic approach of the CUF.•Study of beam-like and plate-like structures with non-orthogonal geometries.
The paper considers quadrature-cubic interpolation in problems of restoration of functions of two arguments. The properties of the most common finite elements Q12 (Lagrange version) and Q10 ...(Serendipity version) are investigated as finite elements. For more than fifty years, researchers of the finite element method have known the procedure of converting the Lagrangian model to the serendipity model. But, as is known, not all results when using this procedure satisfy users, especially proponents of physical interpretation. We are talking about the magnitude of the nodal loads of the uniform mass force of the serendipity finite element. Thus, if we consider the finite element Q10, it receives the physical inadequacy of the "spectrum" as an inheritance from the "parent" pair of Lagrangian finite elements Q8 and Q12. In Pascal's scheme, there is also a hidden connection between the finite element Q10 and the finite element Lagrangian Q12. The analysis of the inherited properties suggests that it is fundamentally possible for a Q10 substitute-base to exist in nature with the same local and integral characteristics. It turns out that the search for such a base goes beyond the capabilities of traditional modeling methods. An alternative substitute-basis Q10 was found by nonmatrix condensation of the prototype of element Q10, i.e., by using the Lagrangian model Q12. The universal nature of the non-matrix transformation of Q12 into Q10 opens up the possibility of designing a model series of mixed finite elements with physically adequate spectra of nodal loads.
•A novel overlapping finite element is presented to solve transient wave propagation problems.•The Bathe implicit time integration method is used.•The solution scheme shows monotonic convergence of ...calculated solutions with decreasing time step size.•The solution scheme shows a solution accuracy almost independent of the direction of wave propagation through the mesh.•A dispersion analysis is given and various example problems are solved to illustrate the performance of the solution scheme.
We present novel overlapping finite elements used with the Bathe time integration method to solve transient wave propagation problems. The solution scheme shows two important properties that have been difficult to achieve in the numerical solution of general wave propagations: monotonic convergence of calculated solutions with decreasing time step size and a solution accuracy almost independent of the direction of wave propagation through the mesh. The proposed scheme can be efficiently used with irregular meshes. These properties make the scheme (the combined spatial and temporal discretizations) promising to solve general wave propagation problems in complex geometries involving multiple waves. A dispersion analysis is given and various example problems are solved to illustrate the performance of the solution scheme.
We design and analyze a new adaptive stabilized finite element method. We construct a discrete approximation of the solution in a continuous trial space by minimizing the residual measured in a dual ...norm of a discontinuous test space that has inf–sup stability. We formulate this residual minimization as a stable saddle-point problem, which delivers a stabilized discrete solution and a residual representation that drives the adaptive mesh refinement. Numerical results on an advection–reaction model problem show competitive error reduction rates when compared to discontinuous Galerkin methods on uniformly refined meshes and smooth solutions. Moreover, the technique leads to optimal decay rates for adaptive mesh refinement and solutions having sharp layers.
•We propose a stabilized FEM based on residual minimization of a discrete-dual norm.•Well-posedness is ensured by standard properties of Discontinuous Galerkin methods.•Mesh adaptivity is efficiently driven by a built-in error estimator of the scheme.•Accuracy and robustness of the scheme is confirmed by 2D and 3D advective problems.•Piecewise discontinuous test spaces, while conforming trial spaces, are considered.
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