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Permanent Magnet (PM) type Longitudinal Flux Linear Motors (LFLMs) are electromagnetic devices which can develop directly powerful linear motion. This paper presents statistical optimum design of PM type LFLM to reduce the weight of the machine with the constraints of avg. thrust force using the penalty method with characteristics function and Response Surface Methodology (RSM). The contribution and effect of the each design variable on the characteristic function is evaluated by the analysis of means (ANOM) and optimum design set is determined. The reduced gradient algorithm is adopted in RSM. Therefore, it is expected that the proposed optimization procedure using penalty method and RSM can be easily utilized to solve the optimization problem with constraint of electric machine.

We derive an a posteriori error estimate for an abstract saddle-point problem when a penalty term is added to stabilize the variational formulation, the aim being to optimize the choice of the penalty parameter. As an application, we consider a discretization of the Stokes problem obtained by combining the penalty technique and the finite element method, we perform its a posteriori analysis in a detailed way and present some numerical experiments on adaptive meshes which are in good agreement with the results of the analysis and confirm its interest.

This paper is concerned with regime-switching American option pricing. We develop new numerical schemes by extending the penalty method approach and by employing the θ-method. With regime-switching, American option prices satisfy a system of m free boundary value problems, where m is the number of regimes considered for the market. An (optimal) early exercise boundary is associated with each regime. Straightforward implementation of the θ-method would result in a system of nonlinear equations requiring a time-consuming iterative procedure at each time step. To avoid such complications, we implement an implicit approach by explicitly treating the nonlinear terms and/or the linear terms from other regimes, resulting in computationally efficient algorithms. We establish an upper bound condition for the time step size and prove that under the condition the implicit schemes satisfy a discrete version of the positivity constraint for American option values. We compare the implicit schemes with a tree model that generalizes the Cox-Ross-Rubinstein (CRR) binomial tree model, and with an analytical approximation solution for two-regime case due to Buffington and Elliott. Numerical examples demonstrate the accuracy and stability of the new implicit schemes.

The American option pricing problem is examined in this work using a regime switching finite moment log-stable model. The option prices under this model are governed by a coupled system of fractional partial differential equations. Combination of the coupled system and the spatial-fractional derivative makes it extremely difficult to find an analytic solution. We have constructed a numerical algorithm to numerically solve such problems. The developed predictor-corrector type method is highly efficient and reliable in solving coupled system in each regime having different volatility and interest rates. Two-sided Riesz space fractional diffusion term is approximated using fractional finite difference scheme whereas the classical space derivative term is approximated using central difference formula. Splitting technique is utilized to construct a highly efficient scheme which can also be implemented on parallel processors. Stability and error analysis of the scheme is proved analytically and demonstrated through numerical experiments. Effect of the order of the fractional derivative (also called tail index) on the option prices is shown through graphs by performing numerical experiments for different values of the tail index.

This paper presents a unified framework for dynamic analysis of vehicle-bridge interaction (VBI) systems using a commercial finite element software suite (ABAQUS$\text{®}$). This framework can provide bridge designers and engineering practitioners with a general platform to analyze the coupled system with high modeling efficiency and accuracy in modeling and outputting. Moreover, it has readily available nonlinear material/element models and nonlinear dynamic analysis functions for complex structures. This analysis framework was first validated with a classical VBI problem involving a sprung mass moving on a simply supported beam, whose closed-form solution is readily available. Validation for the application on complex structure was then presented with a typical 16-car Japanese high-speed train (Shinkansen) and a three-block bridge. The cars comprised car bodies, bogies and wheelsets, which were all modeled as rigid bodies and which were connected with springs and dashpots. The bridge was modeled with typical three-dimensional solid elements. Interaction between wheelsets and tracks was realized using the penalty method. Rail irregularity was also considered in the analysis. The consistency between calculated dynamic responses and field experiment data of certain pre-specified observation points validated the proposed method. Furthermore, ease in analyzing VBI problems involving nonlinear material properties and with high spatial resolutions was demonstrated with a classical cracked beam problem: a point mass moving on a simply supported cracked beam. Both linear and nonlinear crack models were employed. The former model assigned crack surfaces with a mechanical contact property and showed its accuracy in comparison to the reference model. The latter assigned a nonlinear material model in crack-prone zones and illustrated the potential applicability to dynamic crack propagation simulation in VBI problems. The present framework was further applied to seismic response analysis of a train-bridge interaction system involving material nonlinearity and separation between track and wheel under a strong earthquake.

A marine riser operated in a deep-water field could be substantially affected by large amounts of movement of the floating platform, which is more complicated and very challenging to analyze. This paper presents a mathematical model involving nonlinear dynamic response analysis of a marine riser caused by sways and heave motions at the top end, which are treated as the constraint conditions. The nonlinear equation of motion, arising from the nonlinearity of the ocean current and wave loadings, is derived and written in general matrix form using the finite element method. The excitation caused by platform movement is imposed on the riser system through the time-dependent constrained condition using the penalty method. The advantages of this method are that it is easily implemented on the nonlinear equation of motion and it requires no additional unknown variable, and thus consumes less computational time. By this method, the stiffness matrix and the force vector of the system are then modified, enforcing top-end vessel motion. The dynamic responses are evaluated by using numerical time integration based on Newmark’s method with direct iteration. The effects of the oscillation frequency of top-end vessel sway and heave motions on the nonlinear dynamic characteristics of the riser are investigated. The numerical results reveal that the riser responses to the top-end vessel excitation behave like a periodic motion, which is conformable to the characteristics of vessel movements. The increase in the oscillation frequency of the top-end vessel increases the maximum displacement amplitude for both the horizontal and vertical directions. The directional motion of the vessel also significantly influences the response amplitude of the riser.

This study provides an overview on the effect of porosity on free vibrations and more importantly aeroelastic stability margins of cylindrical shells. A general formulation for cylindrical shells is first developed including the effects of shear deformation and rotary inertia along with Sander’s rigid body rotation modification. Two porosity distributions of even and uneven are considered for functionally graded shells. The most general form of power-law model which is known as four-parameter power-law is utilized to provide a clear understanding for the qualification of functionally graded material. A Ritz-based solution algorithm being capable of representing all combinations of symmetric and asymmetric boundary conditions by a penalty method is also introduced. In addition to the capability of satisfying all boundary conditions, the presented solution method is very fast in terms of convergence and computational effort. Various parametric studies are provided and practical results are reported.

We study a new mathematical model which describes the equilibrium of a locking material in contact with a foundation. The contact is frictionless and is modeled with a nonsmooth multivalued interface law which involves unilateral constraints and subdifferential conditions. We describe the model and derive its weak formulation, which is in the form of an elliptic variational–hemivariational inequality for the displacement field. Then, we establish the existence of a unique weak solution to the problem. Next, we introduce a penalty method, for which we state and prove a convergence result. Finally, we consider a particular version of the model for which we prove the continuous dependence of the solution on the bounds which govern the locking and the normal displacement constraints, respectively. We apply this convergence result in the study of an optimization problem associated to the contact model.

In this paper, we present a three-dimensional discrete model governing the deformation of a viral capsid, modelled as a regular icosahedron and subjected not to cross a given flat rigid surface on which it initially lies in correspondence of one vertex only. First, we set up the model in the form of a set of variational inequalities posed over a non-empty, closed and convex subset of a suitable space. Second, we show the existence and uniqueness of the solution for the proposed model. Third, we numerically test this model and we observe that the outputs of the numerical experiments comply with physics. Finally, we establish the existence of solutions for the corresponding time-dependent version of the obstacle problem under consideration.

In this paper we present an rp-adaptive discretization strategy to perform unilateral two-dimensional (2D) mechanical contact simulations by combining the r- and p-versions of the finite element method (FEM). The p-version leaves the finite element mesh unchanged and increases the shape function's polynomial degree in order to obtain convergence toward the exact solution of the underlying mathematical model. The r-method relocates nodes of an existing FE-mesh to improve the discretization of a given problem without introducing additional degrees of freedom, therefore, keeping the problem size fixed. The rp-version, which is a combination of the two aforementioned methods, is used in our study to move a node of the FE-mesh to the end of the contact zone to account for the loss of regularity that arises due to the change from contact to noncontact along the edge. It will be shown that highly accurate results can be obtained by using high-order (p) finite elements in combination with the penalty method and a relocation (r) of element nodes.

A freely combined three-dimensional (3D) discrete and finite element (FE) algorithm is developed. The algorithm, which is based on the method combining the discrete element (DE) and the corresponding node of the FE proposed by Lei and Zang, treats each combined point on the facet of the FE as the corresponding node of the method. This new algorithm includes global search, local search and the combined force calculation processes. The combined force between the DE and FE is processed by using a penalty function method. Following that, the corresponding numerical code is implemented into the in-house developed code, a dynamic explicit DE/FE code named CDFP. To test the accuracy of the proposed algorithm and the corresponding numerical code, the vibration process of two beams under dynamic force and the vibration process of two glass plates under impact of rigid sphere are simulated in elastic range. By comparing the results with those calculated by using LS-DYNA, it is shown that they agree with each other very well.

This paper presents a free vibration analysis of cable structures based on the isogeometric approach. The nonuniform rational B-splines (NURBS) basis functions are employed to represent both the exact geometry of cable and displacement fields. In order to enrich the basis functions, the $h$-, $p$- and $k$-refinement strategies are implemented. Therefore, these refinement schemes increase the accuracy of solution fields. For determining the static configuration of slack cables as a reference configuration, the well-known penalty method is used. Three numerical examples for slack and taut cable structures are presented in which different refinement schemes are utilized to obtain the converged results. The accuracy and reliability of the present numerical method are verified by comparing the natural frequencies with the results given by other researchers.

By using the improved moving least-square (IMLS) approximation to present the shape function, the improved element-free Galerkin (IEFG) method is investigated to solve diffusional drug release problems in this paper. In order to get the discretized equation system, Galerkin weak form of a diffusional drug release problem is used with applying essential boundary conditions using the penalty method. The difference method is applied for discretization of time domain. Then the formulae of IEFG method for solving diffusional drug release problems are presented. Three numerical example problems are given to study the convergence of solutions of IEFG method in this paper. The influences of scale parameters of influence domain, penalty factor and node distribution on the accuracy of the solutions of IEFG method are discussed. Compared with finite element method, the correctness of IEFG method in this paper is shown.

In this study, the dynamical contact simulation by the finite element method (FEM) for the chemical mechanical polishing (CMP) process is presented. In order to simulate CMP, a numerical model for surface roughness layer of the polishing pad and a numerical procedure for FE contact analysis are newly presented. In the analysis model, the surface roughness layer of the polishing pad is assumed as a flat soft layer. The elastic modulus of the soft layer is treated as a fitting parameter between the experimental results and the numerical model, while the patterned wafer can be assumed as a rigid body. The distribution of contact pressure between the patterned wafer and the polishing pad is computed by FEM. The pattern topography is modified according to the pressure dependency of the polishing rate. The iterations of this numerical procedure and the topography modification give the progress of the polishing process dynamically. In order to solve the contact problems, the finite element discretization needs the minimization of total potential energy with gap-contact constraint conditions. For the practical dynamical simulation, the modified stiffness matrix with the penalty numbers is introduced, Although this numerical procedure is based on the linear contact theory, the solution process with iterative schemes is required for getting a stable contact status. Finally, several 2-D numerical analyses for CMP process are demonstrated. Compared with the experiment, the results by the proposed numerical simulation agree with the experimental results very well and the validity of the proposed numerical model for CMP process is clarified.