Penalty Technique Research Articles

A stabilized cut discontinuous Galerkin framework for elliptic boundary value and interface problems

We develop a stabilized cut discontinuous Galerkin framework for the numerical solution of elliptic boundary value and interface problems on complicated domains. The domain of interest is embedded in a structured, unfitted background mesh in Rd, so that the boundary or interface can cut through it in an arbitrary fashion. The method is based on an unfitted variant of the classical symmetric interior penalty method using piecewise discontinuous polynomials defined on the background mesh. Instead of the cell agglomeration technique commonly used in previously introduced unfitted discontinuous Galerkin methods, we employ and extend ghost penalty techniques from recently developed continuous cut finite element methods, which allows for a minimal extension of existing fitted discontinuous Galerkin software to handle unfitted geometries. Identifying four abstract assumptions on the ghost penalty, we derive geometrically robust a priori error and condition number estimates for the Poisson boundary value problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. We also demonstrate how the framework can be elegantly applied to discretize high contrast interface problems. The theoretical results are illustrated by a number of numerical experiments for various approximation orders and for two and three-dimensional test problems.

Buckling and post-buckling analysis of cracked stiffened panels via an X-Ritz method

A multi-domain eXtended Ritz formulation, called X-Ritz, for the analysis of buckling and post-buckling of stiffened panels with cracks is presented. The theoretical framework is based on the First-order Shear Deformation Theory and accounts for von Kármán's geometric nonlinearities. The structure is modeled as assembly of plate elements. Penalty techniques are used to fulfill the continuity condition along the edges of contiguous elements and to satisfy essential boundary conditions requirements. The use of an extended set of approximating functions allows to model through-the-thickness cracks and to capture the crack opening and tip singular fields as well as the structural behavior within each single domain. Numerical results for buckling and post-buckling of cracked stiffened panels are compared with finite elements simulations and literature solutions, showing the accuracy and potential of the proposed approach.

Gauss–Seidel Method for Multi-leader–follower Games

The multi-leader–follower game has many applications such as the bilevel structured market in which two or more enterprises, called leaders, have initiatives, and the other firms, called followers, observe the leaders’ decisions and then decide their own strategies. A special case of the game is the Stackelberg model, or the single-leader–follower game, which has been studied for many years. The Stackelberg game may be reformulated as a mathematical program with equilibrium constraints, which has also been studied extensively in recent years. On the other hand, the multi-leader–follower game may be formulated as an equilibrium problem with equilibrium constraints, in which each leader’s problem is an mathematical program with equilibrium constraints. However, finding an equilibrium point of an equilibrium problem with equilibrium constraints is much more difficult than solving a single mathematical program with equilibrium constraints, because each leader’s problem contains those variables which are common to other players’ problems. Moreover, the constraints of each leader’s problem depend on the other rival leaders’ strategies. In this paper, we propose a Gauss–Seidel type algorithm with a penalty technique for solving an equilibrium problem with equilibrium constraints associated with the multi-leader–follower game, and then suggest a refinement procedure to obtain more accurate solutions. We discuss convergence of the algorithm and report some numerical results to illustrate the behavior of the algorithm.

A probability-based coevolving multi-objective algorithm for antenna array synthesis

The synthesis of the antenna is a multi-objective problem which contains large-scale decision variables including but not limited to the elements’ locations, amplitudes, and phases. Numerous researches have introduced heuristic algorithms on the synthesis optimization, and some of them employed multi-objective strategies which are proved to provide better optimal solutions for deciders. Moreover, since more complex relationships and a larger number of decision variables have appeared in nowadays array antennas, the study simultaneously considers high-dimensional decision space, as well as the multiple requirements is essential for highly directive patterns. However, no existing literature can be found in the corresponding field to the best of our knowledge. In this paper, a probability-based coevolving particle swarm optimization (PCMOPSO) is proposed for multi-objective optimization. In PCMOPSO, decision variables are allocated into subgroups based on the cooperative coevolution framework, and then optimized through the probability-based learning strategy to accelerate convergence simultaneously with maintaining diversity. Meanwhile, a grouping penalty (GP) technique is proposed to improve the grouping technique in PCMOPSO. The proposed algorithm has been tested for efficacy on several benchmark functions under different decision dimensions. The experimental results exhibit that PCMOPSO has superior performance relative to other similar methods. Furthermore, PCMOPSO is applied to optimize the array parameters under several antenna synthesis environments, including large array design cases. Significant improvement was obtained compared to other state-of-the-art multi-objective algorithms and known optimal solutions in the literature.

Rotation‐free triangular shell element using node‐based smoothed finite element method

SummaryIn this paper, a triangular thin flat shell element without rotation degrees of freedom is proposed. In the Kirchhoff hypothesis, the first derivative of the displacement must be continuous because there are second‐order differential terms of the displacement in the weak form of the governing equations. The displacement is expressed as a linear function and the nodal rotation is defined using node‐based smoothed finite element method. The rotation field is approximated using the nodal rotation and linear shape functions. This rotation field is linear in an element and continuous between elements. The curvature is defined by differentiating the rotation field, and the stiffness is calculated from the curvature. A hybrid stress triangular membrane element was used to construct the shell element. The penalty technique was used to apply the rotation boundary conditions. The proposed element was verified through several numerical examples.

A variational approach to vibrations of laminated composite plates with a line hinge

A Ritz variational approach for the analysis of free vibration of plates with internal hinges distributed along a straight line is presented. The plate is modeled with the first order shear deformation theory and symmetric stacking sequences. The line with hinges is modeled by using subdomain decomposition of the plate coupled with penalty techniques, which causes an increment of the variational statement to assure the continuity conditions along the contact boundary of the plate's subdomains. In order to obtain an indication of the accuracy of the developed mathematical model and the proposed technique, some cases available in the literature have been considered. Original results are presented for solution of representative symmetric multilayered plates showing the effects of the line hinge parameters on the natural frequencies and mode shapes.

Simulation of acoustic and flexural-gravity waves in ice-covered oceans

We introduce a provably stable, high-order-accurate finite difference method for simulation of acoustic and flexural gravity waves in compressible, inviscid fluids partially covered by a thin elastic layer. Such waves arise when studying ocean wave interactions with floating ice shelves, sea ice, and floating structures. Particular emphasis is on a well-posed interface treatment of the fluid-ice coupling. To ensure numerical stability and efficiency, finite difference approximations based on the summation-by-parts (SBP) framework are combined with a penalty technique (simultaneous approximation term, SAT) to impose the boundary and interface conditions. The resulting SBP-SAT approximations are time integrated with an unconditionally stable finite difference method. Numerical simulations in 2D corroborate the predicted efficiency and stability behaviors. The method can be used in its current form to study transmission of ocean waves and tsunamis through ice shelves, and upon coupling to an elastic half-space beneath the ice and water, to study ice motions associated with long-period seismic surface waves.

Differential evolution with the adaptive penalty method for structural multi-objective optimization

Real-world engineering design problems, like structural optimization, can be characterized as a multi-objective optimization when two or more conflicting objectives are in the problem formulation. The differential evolution (DE) algorithm is nowadays one of the most popular meta-heuristics to solve optimization problems in continuous search spaces and has attracted much attention in multi-objective optimization due to its simple implementation and efficiency when solving real-world problems. A recent paper has shown that GDE3, a well-known DE-based algorithm, performs efficiently when solving structural multi-objective optimization problems. Also an adaptive penalty technique called APM was adopted to handle constraints. However, the authors did not investigate the contribution of this technique and that of the GDE3 algorithm separately. So, in this work, the results obtained by GDE3 equipped with the APM scheme (denoted here by GDE3 + APM) are compared with those found by the original GDE3 in order to investigate the advantages and limitations of this constraint handling technique in those problems. The results of the GDE3 + APM are also compared with the most commonly used multi-objective meta-heuristic, namely NSGA-II, in order to comparatively evaluate the quality of the solutions obtained with respect to other algorithms from the literature. The analysis indicates that GDE3 + APM is more efficient than both GDE3 and NSGA-II in most performance metrics used when solving the structural multi-objective optimization problems considered here, suggesting that the GDE3 + APM algorithm is promising in this area, and that the APM technique makes a considerable contribution to its performance.

High-fidelity Sound Propagation in a Varying 3D Atmosphere

A stable and high-order accurate upwind finite difference discretization of the 3D linearized Euler equations is presented. The discretization allows point sources, a varying atmosphere and curved topography. The advective terms are discretized using recently published upwind summation-by-parts (SBP) operators and the boundary conditions are imposed using a penalty technique. The resulting discretization leads to an explicit ODE system. The accuracy and stability properties are verified for a linear hyperbolic problems in 1D, and for the 3D linearized Euler equations. The usage of upwind SBP operators leads to robust and accurate numerical approximations in the presence of point sources and naturally avoids the onset of spurious oscillations.

A non-linear approximation to the distribution of total expenditure distribution of cruise tourists in Uruguay

This study contributes to the literature on the determinants of tourism spending on cruises at a microeconomic level, through the application of innovative methodologies framed within the machine learning literature. The objective is to study the distribution of the total expenditure of cruise passengers in Uruguay, using data of the 2016–2017 cruise season survey (collected by the Ministry of Tourism of Uruguay). Due to the nature of this variable, we implement a two stages modeling strategy. In the first stage, we model the probability of spending, and in the second, the strictly positive spending. The paper analyze the distribution of conditional expenditure to a set of sociodemographic, travel, contextual and satisfaction variables applying non-linear regression techniques with Lasso penalty and nonparametric techniques such as Random Forest. The empirical results show that the key variables that determine the average spending of cruise tourists are their residence and the port of arrival of the cruise. The analysis of the predictive performance of the models (applied through a training sample and a test sample) shows that Random Forest method has the greater predictive capacity. Finally, the importance variable is analyzed by Random Forest.

A framelet algorithm for de-blurring images corrupted by multiplicative noise

Abstract This paper considers a variational model for restoring images from blurry and speckled observations. This model utilizes the favorable properties of framelet regularization (e.g., the sparsity and multiresolution properties of the framelet) that are well suited for speckle noise reduction. For solving the model, we first propose an approximation model that is motivated by the well-known variable-splitting and penalty techniques in optimization. We then develop an alternating minimization algorithm to solve the approximation model. We also show that the sequence generated by the algorithm converges to the solution of the proposed model. The numerical results on simulated data and real utrasound images demonstrate that our approach outperforms several state-of-the-art algorithms.

Alternative Constraint Handling Technique for Four-Bar Linkage Path Generation

This paper proposes an extension of a new concept for path generation from our previous work by adding a new constraint handling technique. The propose technique was initially designed for problems without prescribed timing by avoiding the timing constraint, while remain constraints are solving with a new constraint handling technique. The technique is one kind of penalty technique. The comparative study is optimisation of path generation problems are solved using self-adaptive population size teaching-learning based optimization (SAP-TLBO) and original TLBO. In this study, two traditional path generation test problem are used to test the proposed technique. The results show that the new technique can be applied with the path generation problem without prescribed timing and gives better results than the previous technique. Furthermore, SAP-TLBO outperforms the original one.

A combined finite element method for elliptic problems posted in domains with rough boundaries

A combined finite element method is presented in this paper to solve the elliptic problems posted in domains with rough boundaries. Solving these problems numerically is difficult because resolving the boundaries usually requires very fine meshes, while good quality meshes often over-refine unnecessarily the interior of the domain. The basic idea of the proposed method is to use a fine mesh with size h in the vicinity of oscillating boundaries and a coarse mesh with size H ≫ h for other portions of the domain to reduce some unnecessary computational effort. The transmission conditions across the fine-coarse mesh interface are treated by the penalty technique. The key point of the method lies in the new scheme employing a weighted average in the definition of the bilinear form, which avoids the affection of the ratio H ∕ h in the error estimate. We prove a quasi-optimal convergence in terms of elements since there is no whole H 2 regularity in the domain with rough boundaries. Numerical results are provided for elliptic equations in domains with non-oscillating or oscillating boundaries to illustrate the theoretical results.

Domain decomposition technique to simulate crack in nonlinear analysis of initially imperfect laminates

In this research, an effective computational technique is carried out for nonlinear and post-buckling analyses of cracked imperfect composite plates. The laminated plates are assumed to be moderately thick so that the analysis can be carried out based on the first-order shear deformation theory. Geometric non-linearity is introduced in the way of von-Karman assumptions for the strain-displacement equations. The Ritz technique is applied using Legendre polynomials for the primary variable approximations. The crack is modeled by partitioning the entire domain of the plates into several sub-plates and therefore the plate decomposition technique is implemented in this research. The penalty technique is used for imposing the interface continuity between the sub-plates. Different out-of-plane essential boundary conditions such as clamp, simply support or free conditions will be assumed in this research by defining the relevant displacement functions. For in-plane boundary conditions, lateral expansions of the unloaded edges are completely free while the loaded edges are assumed to move straight but restricted to move laterally. With the formulation presented here, the plates can be subjected to biaxial compressive loads, therefore a sensitivity analysis is performed with respect to the applied load direction, along the parallel or perpendicular to the crack axis. The integrals of potential energy are numerically computed using Gauss-Lobatto quadrature formulas to get adequate accuracy. Then, the obtained non-linear system of equations is solved by the Newton-Raphson method. Finally, the results are presented to show the influence of crack length, various locations of crack, load direction, boundary conditions and different values of initial imperfection on nonlinear and post-buckling behavior of laminates.

Modelling low Reynolds number vortex-induced vibration problems with a fixed mesh fluid-solid interaction formulation

The present work reports a fluid-rigid solid interaction formulation described within the framework of a fixed-mesh technique. The numerical analysis is focussed on the study of a vortex-induced vibration (VIV) of a circular cylinder at low Reynolds number. The proposed numerical scheme encompasses the fluid dynamics computation in an Eulerian domain where the body is embedded using a collection of markers to describe its shape, and the rigid solid’s motion is obtained with the well-known Newton’s law. The body’s velocity is imposed on the fluid domain through a penalty technique on the embedded fluid-solid interface. The fluid tractions acting on the solid are computed from the fluid dynamic solution of the flow around the body. The resulting forces are considered to solve the solid motion. The numerical code is validated by contrasting the obtained results with those reported in the literature using different approaches for simulating the flow past a fixed circular cylinder as a benchmark problem. Moreo...

A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels

A Rayleigh-Ritz solution approach for generally restrained multilayered variable angle tow stiffened plates in postbuckling regime is presented. The plate model is based on the first order shear deformation theory and accounts for geometrical nonlinearity through the von Kármán’s assumptions. Stiffened plates are modelled as assembly of plate-like elements and penalty techniques are used to join the elements in the assembled structure and to apply the kinematical boundary conditions. General symmetric and unsymmetric stacking sequences are considered and Legendre orthogonal polynomials are employed to build the trial functions. A computer code was developed to implement the proposed approach and to establish its applicability and its features for investigating variable angle tow structures. The proposed solution is validated by comparison with literature and finite elements results. Original results are presented for postbuckling of variable angle tow stiffened plates showing the potentialities of the method.

Noise reduction in Lidar signal using correlation-based EMD combined with soft thresholding and roughness penalty

Empirical mode decomposition (EMD) is widely used to analyze the non-linear and non-stationary signals for noise reduction. In this study, a novel EMD-based denoising method, referred to as EMD with soft thresholding and roughness penalty (EMD-STRP), is proposed for the Lidar signal denoising. With the proposed method, the relevant and irrelevant intrinsic mode functions are first distinguished via a correlation coefficient. Then, the soft thresholding technique is applied to the irrelevant modes, and the roughness penalty technique is applied to the relevant modes to extract as much information as possible. The effectiveness of the proposed method was evaluated using three typical signals contaminated by white Gaussian noise. The denoising performance was then compared to the denoising capabilities of other techniques, such as correlation-based EMD partial reconstruction, correlation-based EMD hard thresholding, and wavelet transform. The use of EMD-STRP on the measured Lidar signal resulted in the noise being efficiently suppressed, with an improved signal to noise ratio of 22.25 dB and an extended detection range of 11 km.

Compatible diagonal-norm staggered and upwind SBP operators

The main motivation with the present study is to achieve a provably stable high-order accurate finite difference discretisation of linear first-order hyperbolic problems on a staggered grid. The use of a staggered grid makes it non-trivial to discretise advective terms. To overcome this difficulty we discretise the advective terms using upwind Summation-By-Parts (SBP) operators, while the remaining terms are discretised using staggered SBP operators. The upwind and staggered SBP operators (for each order of accuracy) are compatible, here meaning that they are based on the same diagonal norms, allowing for energy estimates to be formulated. The boundary conditions are imposed using a penalty (SAT) technique, to guarantee linear stability. The resulting SBP-SAT approximations lead to fully explicit ODE systems. The accuracy and stability properties are demonstrated for linear hyperbolic problems in 1D, and for the 2D linearised Euler equations with constant background flow. The newly derived upwind and staggered SBP operators lead to significantly more accurate numerical approximations, compared with the exclusive usage of (previously derived) central-difference first derivative SBP operators.

Robust full‐wave Maxwell solver in time‐domain using magnetic vector potential with edge elements

Magnetic vector potential (MVP) formulations are widely used in low-frequency eddy-current computation. For full-wave Maxwell problems with coupled resistive, inductive and capacitive effects, existing MVP formulations are either ungauged or with too complicated stabilisation procedure or with non-symmetric system matrix. Since sophisticated preconditioners are necessary to ensure good convergence rates of iterative solvers when used to solve the resultant linear system, it is also important to investigate symmetric gauged formulations which can be solved by more robust state-of-the-art sparse direct solvers. Traditional Coulomb gauged formulation using penalty technique is not feasible for edge elements since the divergence of the edge element basis function is zero within each element. In this study, a novel MVP formulation with Coulomb Gauge for full-wave Maxwell problems using edge elements is proposed. The proposed formulation is symmetric and stable for all frequencies, easy to implement and uniquely solvable. Numerical results obtained using the proposed MVP formulation are demonstrated to showcase its stability and accuracy. The method is very promising in further engineering applications.

Group-Constrained Maximum Correntropy Criterion Algorithms for Estimating Sparse Mix-Noised Channels

A group-constrained maximum correntropy criterion (GC-MCC) algorithm is proposed on the basis of the compressive sensing (CS) concept and zero attracting (ZA) techniques and its estimating behavior is verified over sparse multi-path channels. The proposed algorithm is implemented by exerting different norm penalties on the two grouped channel coefficients to improve the channel estimation performance in a mixed noise environment. As a result, a zero attraction term is obtained from the expected l 0 and l 1 penalty techniques. Furthermore, a reweighting factor is adopted and incorporated into the zero-attraction term of the GC-MCC algorithm which is denoted as the reweighted GC-MCC (RGC-MMC) algorithm to enhance the estimation performance. Both the GC-MCC and RGC-MCC algorithms are developed to exploit well the inherent sparseness properties of the sparse multi-path channels due to the expected zero-attraction terms in their iterations. The channel estimation behaviors are discussed and analyzed over sparse channels in mixed Gaussian noise environments. The computer simulation results show that the estimated steady-state error is smaller and the convergence is faster than those of the previously reported MCC and sparse MCC algorithms.