Interior Penalty Method Research Articles

Formula omitted] interior penalty methods for a dynamic nonlinear beam model

In this work, we aim to develop efficient numerical schemes for a nonlinear fourth-order partial differential equation arising from the so-called dynamic Gao beam model. We use C0 interior penalty finite element methods over the spatial domain to set up the semi-discrete formulations. Convergence results for the semi-discrete case are shown, based on a truncated variational formulation and its equivalent abstract formulations. We combine time discretizations to derive fully discrete numerical formulations. Newton’s method is applied to compute one time step numerical solutions of a nonlinear system. Two numerical examples are provided: one supports our theoretical results and the other presents a buckling state of the Gao beams.

Performance analysis and path planning for UAVs swarms based on RSS measurements

In the localization estimation system, it is well known that the sensor-emitter geometry can seriously impact the accuracy of the location estimate. In this paper, we analyze the optimal deployment for received signal strength (RSS) localization with the measurement noise is set to be distance dependent. First, the Cramer–Rao low bound (CRLB) with distance-dependent noise in RSS localization is calculated and chosen to be the optimality criterion. The optimal deployment is analyzed via angle and distance criterion, respectively. Then, the analytic solutions to the optimal deployment are derived in both static and movable target scenarios. Finally, we extend our work to the path planning problem with constraints and interior penalty method is applied to settle the constrained nonlinear optimization problem. The simulation results show that the path optimization verifies the accuracy of the analytical findings.

An adaptive C0IPG method for the Helmholtz transmission eigenvalue problem

The interior penalty methods using $C^0$ Lagrange elements ($C^0$IPG) developed in the last decade for the fourth order problems are an interesting topic in academia at present. In this paper, we discuss the adaptive fashion of $C^0$IPG method for the Helmholtz transmission eigenvalue problem.We give the a posteriori error indicators for primal and dual eigenfunctions, and prove their reliability and efficiency. We also give the a posteriori error indicator for eigenvalues and design a $C^0$IPG adaptive algorithm. Numerical experiments show that this algorithm is efficient and can get the optimal convergence rate.

Robust and efficient discontinuous Galerkin methods for under-resolved turbulent incompressible flows

We present a robust and accurate discretization approach for incompressible turbulent flows based on high-order discontinuous Galerkin methods. The DG discretization of the incompressible Navier–Stokes equations uses the local Lax–Friedrichs flux for the convective term, the symmetric interior penalty method for the viscous term, and central fluxes for the velocity–pressure coupling terms. Stability of the discretization approach for under-resolved, turbulent flow problems is realized by a purely numerical stabilization approach. Consistent penalty terms that enforce the incompressibility constraint as well as inter-element continuity of the velocity field in a weak sense render the numerical method a robust discretization scheme in the under-resolved regime. The penalty parameters are derived by means of dimensional analysis using penalty factors of order 1. Applying these penalty terms in a postprocessing step leads to an efficient solution algorithm for turbulent flows. The proposed approach is applicable independently of the solution strategy used to solve the incompressible Navier–Stokes equations, i.e., it can be used for both projection-type solution methods as well as monolithic solution approaches. Since our approach is based on consistent penalty terms, it is by definition generic and provides optimal rates of convergence when applied to laminar flow problems. Robustness and accuracy are verified for the Orr–Sommerfeld stability problem, the Taylor–Green vortex problem, and turbulent channel flow. Moreover, the accuracy of high-order discretizations as compared to low-order discretizations is investigated for these flow problems. A comparison to state-of-the-art computational approaches for large-eddy simulation indicates that the proposed methods are highly attractive components for turbulent flow solvers.

Effects of Probability Function on the Performance of Stochastic Programming

Stochastic programming is a valuable optimization tool where used when some or all of the design parameters of an optimization problem are defined by stochastic variables rather than by deterministic quantities. Depending on the nature of equations involved in the problem, a stochastic optimization problem is called a stochastic linear or nonlinear programming problem. In this paper,a stochastic optimization problem is transformed intoan equivalent deterministic problem,which can be solved byany known classical methods (interior penalty method is applied here).The paper mainly focuseson investigatingthe effect of applying various probability functions distributions(normal, gamma, and exponential) for design variables. The following basic required equations to solve nonlinear stochastic problems with various probability functionsfor random variables are derived and sensitivity analyses to studythe effects of distribution function typesand input parameterson the optimum solution are presented as graphs and in tables by studyingtwoconsidered test problems. It is concluded that thedifference between probabilistic and deterministic solutions toa problem, when the normal distribution ofrandom variables isused, is very different fromthe results when gamma and exponential distribution functions are used. Finally, it is shownthat the rate of solution convergence tothe normal distribution is faster than the other distributions.

A priori and a posteriori error control of discontinuous Galerkin finite element methods for the von Kármán equations

This paper analyses discontinuous Galerkin finite element methods (DGFEM) to approximate a regular solution to the von Karman equations defined on a polygonal domain. A discrete inf–sup condition sufficient for the stability of the discontinuous Galerkin discretization of a well-posed linear problem is established, and this allows the proof of local existence and uniqueness of a discrete solution to the nonlinear problem with a Banach fixed point theorem. The Newton scheme is locally second-order convergent and appears to be a robust solution strategy up to machine precision. A comprehensive a priori and a posteriori energy-norm error analysis relies on one sufficiently large stabilization parameter and sufficiently fine triangulations. In case the other stabilization parameter degenerates towards infinity, the DGFEM reduces to a novel C0-interior penalty method (IPDG). In contrast to the known C0-IPDG dueto Brenner et al., (2016, A C0 interior penalty method for a von Karman plate. Numer. Math., 1–30), the overall discrete formulation maintains symmetry of the trilinear form in the first two components—despite the general nonsymmetry of the discrete nonlinear problems. Moreover, a reliable and efficient a posteriori error analysis immediately follows for the DGFEM of this paper, while the different norms in the known C0-IPDG lead to complications with some nonresidual-type remaining terms. Numerical experiments confirm the best-approximation results and the equivalence of the error and the error estimators. A related adaptive mesh-refining algorithm leads to optimal empirical convergence rates for a nonconvex domain.

Supercloseness of Continuous Interior Penalty Methods on Shishkin Triangular Meshes and Hybrid Meshes for Singularly Perturbed Problems with Characteristic Layers

A singularly perturbed convection–diffusion problem posed on the unit square is solved using a continuous interior penalty (CIP) method. The mesh used is a Shishkin triangular mesh or a Shishkin hybrid mesh consisting of triangles and rectangles. For the CIP method, a variant of Oswald interpolation operator is introduced for a discrete inf-sup stability, which is proved in a new norm stronger than the the usual CIP norm. This stability and a new cancellation technique enable new supercloseness results for the CIP method: the computed solutions on the triangular mesh and the hybrid mesh are shown to be 3/2 order and 2 order (up to a logarithmic factor) convergent in the new norm to the interpolants of the true solution, respectively. These convergence orders are uniformly valid with respect to the diffusion parameter and imply that for the Shishkin mesh the hybrid mesh is superior to the triangular one. Numerical experiments illustrate these theoretical results.

Stabilized equal low-order finite elements in ice sheet modeling \u2013 accuracy and robustness

We investigate the accuracy and robustness of one of the most common methods used in glaciology for finite element discretization of the \U0001d52d-Stokes equations: linear equal order finite elements with Galerkin least-squares (GLS) stabilization on anisotropic meshes. Furthermore, we compare the results to other stabilized methods. We find that the vertical velocity component is more sensitive to the choice of GLS stabilization parameter than horizontal velocity. Additionally, the accuracy of the vertical velocity component is especially important since errors in this component can cause ice surface instabilities and propagate into future ice volume predictions. If the element cell size is set to the minimum edge length and the stabilization parameter is allowed to vary non-linearly with viscosity, the GLS stabilization parameter found in literature is a good choice on simple domains. However, near ice margins the standard parameter choice may result in significant oscillations in the vertical component of the surface velocity. For these reasons, other stabilization techniques, in particular the interior penalty method, result in better accuracy and are less sensitive to the choice of stabilization parameter. During this work, we also discovered that the manufactured solutions often used to evaluate errors in glaciology are not reliable due to high artificial surface forces at singularities. We perform our numerical experiments in both FEniCS and Elmer/Ice.

Quasi-Optimal Nonconforming Methods for Symmetric Elliptic Problems. III---Discontinuous Galerkin and Other Interior Penalty Methods

We devise new variants of the following nonconforming finite element methods: discontinuous Galerkin methods of fixed arbitrary order for the Poisson problem, the Crouzeix--Raviart interior penalty method for linear elasticity, and the quadratic $C^0$ interior penalty method for the biharmonic problem. Each variant differs from the original method only in the discretization of the right-hand side. Before applying the load functional, a linear operator transforms nonconforming discrete test functions into conforming functions such that stability and consistency are improved. The new variants are thus quasi-optimal with respect to an extension of the energy norm. Furthermore, their quasi-optimality constants are uniformly bounded for shape regular meshes and tend to 1 as the penalty parameter increases.

Residual-Based Variational Multiscale Modeling in a Discontinuous Galerkin Framework

We develop the general form of the variational multiscale method in a discontinuous Galerkin framework. Our method is based on the decomposition of the true solution into discontinuous coarse-scale and discontinuous fine-scale parts. The obtained coarse-scale weak formulation includes two types of fine-scale contributions. The first type corresponds to a fine-scale volumetric term, which we formulate in terms of a residual-based model that also takes into account fine-scale effects at element interfaces. The second type consists of independent fine-scale terms at element interfaces, which we formulate in terms of a new fine-scale “interface model.” We demonstrate for the one-dimensional Poisson problem that existing discontinuous Galerkin formulations, such as the interior penalty method, can be rederived by choosing particular fine-scale interface models. The multiscale formulation thus opens the door for a new perspective on discontinuous Galerkin methods and their numerical properties. This is demonstr...

$C^0$ Interior Penalty Methods for an Elliptic Distributed Optimal Control Problem on Nonconvex Polygonal Domains with Pointwise State Constraints

We design and analyze $C^0$ interior penalty methods for an elliptic distributed optimal control problem on nonconvex polygonal domains with pointwise state constraints.MSC codeselliptic distributed optimal control problemspointwise state constraintsnonconvex domainsvariational inequalitiesdiscontinuous Galerkin methodsMSC codes49J2065K1565N30

Semi-analytical discontinuous Galerkin finite element method for the calculation of dispersion properties of guided waves in plates.

The development of reliable guided waves inspection systems is conditioned by an accurate knowledge of their dispersive properties. The semi-analytical finite element method has been proven to be very practical for modeling wave propagation in arbitrary cross-section waveguides. However, when it comes to computations on complex geometries to a given accuracy, it still has a major drawback: the high consumption of resources. Recently, discontinuous Galerkin finite element method (DG-FEM) has been found advantageous over the standard finite element method when applied as well in the frequency domain. In this work, a high-order method for the computation of Lamb mode characteristics in plates is proposed. The problem is discretised using a class of DG-FEM, namely, the interior penalty methods family. The analytical validation is performed through the homogeneous isotropic case with traction-free boundary conditions. Afterwards, functionally graded material plates are analysed and a numerical example is presented. It was found that the obtained results are in good agreement with those found in the literature.

A Higher-Order Discontinuous Galerkin/Arbitrary Lagrangian Eulerian Partitioned Approach to Solving Fluid–Structure Interaction Problems with Incompressible, Viscous Fluids and Elastic Structures

This manuscript presents a discontinuous Galerkin-based numerical method for solving fluid–structure interaction problems involving incompressible, viscous fluids. The fluid and structure are fully coupled via two sets of coupling conditions. The numerical approach is based on a high-order discontinuous Galerkin (with Interior Penalty) method, which is combined with the Arbitrary Lagrangian–Eulerian approach to deal with the motion of the fluid domain, which is not known a priori. Two strongly coupled partitioned schemes are considered to resolve the interaction between fluid and structure: the Dirichlet–Neumann and the Robin–Neumann schemes. The proposed numerical method is tested on a series of benchmark problems, and is applied to a fluid–structure interaction problem describing the flow of blood in a patient-specific aortic abdominal aneurysm before and after the insertion of a prosthesis known as stent graft. The proposed numerical approach provides sharp resolution of jump discontinuities in the pressure and normal stress across fluid–structure and structure–structure interfaces. It also provides a unified framework for solving fluid–structure interaction problems involving nonlinear structures, which may develop shock wave solutions that can be resolved using a unified discontinuous Galerkin-based approach.

Supercloseness of the continuous interior penalty method for singularly perturbed problems in 1D: Vertex-cell interpolation

A continuous interior penalty method with piecewise polynomials of degree p≥2 is applied on a Shishkin mesh to solve a singularly perturbed convection–diffusion problem, whose solution has a single boundary layer. This method is analyzed by means of a series of integral identities developed for the convection terms. Then we prove a supercloseness bound of order 5/2 for a vertex-cell interpolation when p=2. The sharpness of our analysis is supported by some numerical experiments. Moreover, numerical tests show supercloseness clearly for p≥3.

A stabilized Nitsche cut finite element method for the Oseen problem

We provide the numerical analysis for a Nitsche-based cut finite element formulation for the Oseen problem, which has been originally presented for the incompressible Navier–Stokes equations by Schott and Wall (2014) and allows the boundary of the domain to cut through the elements of an easy-to-generate background mesh. The formulation is based on the continuous interior penalty (CIP) method of Burman et al. (2006) which penalizes jumps of velocity and pressure gradients over inter-element faces to counteract instabilities arising for high local Reynolds numbers and the use of equal order interpolation spaces for the velocity and pressure. Since the mesh does not fit the boundary, Dirichlet boundary conditions are imposed weakly by a stabilized Nitsche-type approach. The addition of CIP-like ghost-penalties in the boundary zone allows to prove that our method is inf–supstable and to derive optimal order a priori error estimates in an energy-type norm, irrespective of how the boundary cuts the underlying mesh. All applied stabilization techniques are developed with particular emphasis on low and high Reynolds numbers. Two- and three-dimensional numerical examples corroborate the theoretical findings. Finally, the proposed method is applied to solve the transient incompressible Navier–Stokes equations on a complex geometry.

A discontinuous Galerkin method for the mono-energetic Fokker–Planck equation based on a spherical interior penalty formulation

We present a new discretization of the mono-energetic Fokker–Planck equation. We build on previous work (Kópházi and Lathouwers, 2015) where we devised an angular discretization for the Boltzmann equation, allowing for both heterogeneous and anisotropic angular refinement. The angular discretization is based on a discontinuous finite element method on the unit sphere. Here we extend the methodology to include the effect of the Fokker–Planck scatter operator describing small angle particle scatter. We describe the construction of an interior penalty method on the sphere surface. Results are provided for a variety of test cases, ranging from purely angular to fully three-dimensional. The results show that the scheme can resolve highly forward-peaked flux distributions with forward-peaked scatter.

An interior penalty method for a large-scale finite-dimensional nonlinear double obstacle problem

• Novel interior penalty formulation for large-scale bounded NCPs in engineering. • Proof of unique solvability of the penalty equation. • Establishment of convergence theory for the penalty method. • Jacobian matrix in Newton method shown to be positive-definite and an M-matrix. • Linearized system in Newton method decomposable into two decoupled subsystems. We propose and analyze an interior penalty method for a finite-dimensional large-scale bounded Nonlinear Complementarity Problem (NCP) arising from the discretization of a differential double obstacle problem in engineering. Our approach is to approximate the bounded NCP by a nonlinear algebraic equation containing a penalty function with a penalty parameter μ > 0. The penalty equation is shown to be uniquely solvable. We also prove that the solution to the penalty equation converges to the exact one at the rate O ( μ 1 / 2 ) as μ → 0. A smooth Newton method is proposed for solving the penalty equation and it is shown that the linearized system is reducible to two decoupled subsystems. Numerical experiments, performed on some non-trivial test examples, demonstrate the computed rate of convergence matches the theoretical one.

A $$C^0$$ Linear Finite Element Method for Biharmonic Problems

In this paper, a $$C^0$$ linear finite element method for biharmonic equations is constructed and analyzed. In our construction, the popular post-processing gradient recovery operators are used to calculate approximately the second order partial derivatives of a $$C^0$$ linear finite element function which do not exist in traditional meaning. The proposed scheme is straightforward and simple. More importantly, it is shown that the numerical solution of the proposed method converges to the exact one with optimal orders both under $$L^2$$ and discrete $$H^2$$ norms, while the recovered numerical gradient converges to the exact one with a superconvergence order. Some novel properties of gradient recovery operators are discovered in the analysis of our method. In several numerical experiments, our theoretical findings are verified and a comparison of the proposed method with the nonconforming Morley element and $$C^0$$ interior penalty method is given.

A Symmetric Direct Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations

AbstractIn this work, we investigate the numerical approximation of the compressible Navier-Stokes equations under the framework of discontinuous Galerkin methods. For discretization of the viscous and heat fluxes, we extend and apply the symmetric direct discontinuous Galerkin (SDDG) method which is originally introduced for scalar diffusion problems. The original compressible Navier-Stokes equations are rewritten into an equivalent form via homogeneity tensors. Then, the numerical diffusive fluxes are constructed from the weak formulation of primal equations directly without converting the second-order equations to a first-order system. Additional numerical flux functions involving the jump of second order derivative of test functions are added to the original direct discontinuous Galerkin (DDG) discretization. A number of numerical tests are carried out to assess the practical performance of the SDDG method for the two dimensional compressible Navier-Stokes equations. These numerical results obtained demonstrate that the SDDG method can achieve the optimal order of accuracy. Especially, compared with the well-established symmetric interior penalty (SIP) method [18], the SDDG method can maintain the expected optimal order of convergence with a smaller penalty coefficient.

A nonconforming scheme for non-Fickian flow in porous media

In this paper, we construct a semi-discrete scheme and a fully discrete scheme using the Wilson nonconforming element for the parabolic integro-differential equation arising in modeling the non-Fickian flow in porous media by the interior penalty method. Without using the conventional elliptic projection, which was an indispensable tool in the convergence analysis of finite element methods in previous literature, we get an optimal error estimate which is only determined by the interpolation error. Finally, we give some numerical experiments to show the efficiency of the method.