Search results

When simulating fluid-structure interaction (FSI), it is often essential that the no-slip condition is accurately enforced at the wetted boundary of the structure. This paper aims to evaluate the relative strengths and limitations of the penalty and Lagrange multiplier methods, within the context of modelling FSI, through a comparative analysis.

In the immersed boundary method, the no-slip condition is typically imposed by augmenting the governing equations of the fluid with an artificial body force. The relative accuracy and computational time of the penalty and Lagrange multiplier formulations of this body force are evaluated by using each to solve three test problems, namely, flow through a channel, the harmonic motion of a cylinder through a stationary fluid and the vortex-induced vibration (VIV) of a cylinder.

The Lagrange multiplier formulation provided an accurate solution, especially when enforcing the no-slip condition, and was robust as it did not require “tuning” of problem specific parameters. However, these benefits came at a higher computational cost relative to the penalty formulation. The penalty formulation achieved similar levels of accuracy to the Lagrange multiplier formulation, but only if the appropriate penalty factor was selected, which was difficult to determine a priori.

Both the Lagrange multiplier and penalty formulations of the immersed boundary method are prominent in the literature. A systematic quantitative comparison of these two methods is presented within the same computational environment. A novel application of the Lagrange multiplier method to the modelling of VIV is also provided.

The purpose of this paper is to further simplify the use of NURBS in industrial environnements. Although isogeometric analysis (IGA) has been the object of intensive studies over the past decade, its massive deployment in industrial analysis still appears quite marginal. This is partly due to its implementation, which is not straightforward with respect to the elementary structure of finite element (FE) codes. This often discourages industrial engineers from adopting isogeometric capabilities in their well-established simulation environment.

Based on the concept of Bézier and Lagrange extractions, a novel method is proposed to implement IGA from an existing industrial FE code with the aim of bringing human implementation effort to the minimal possible level (only using standard input-output of finite element analysis (FEA) codes, avoid code-dependent subroutines implementation). An approximate global link to go from Lagrange polynomials to non-uniform-rational-B-splines functions is formulated, which enables the whole FE routines to be untouched during the implementation.

As a result, only the linear system resolution step is bypassed: the resolution is performed in an external script after projecting the FE system onto the reduced, more regular and isogeometric basis. The novel procedure is successfully validated through different numerical experiments involving linear and nonlinear isogeometric analyses using the standard input/output of the industrial FE software Code_Aster.

A non-invasive implementation of IGA into FEA software is proposed. The whole FE routines are untouched during the novel implementation procedure; a focus is made on the IGA solution of nonlinear problems from existing FEA software; technical details on the approach are provided by means of illustrative examples and step-by-step implementation; the methodology is evaluated on a range of two- and three-dimensional elasticity and elastoplasticity benchmarks solved using the commercial software Code_Aster.

This meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only the high wavenumber problems, but also the variable wave number problems.

In this paper, the authors developed a meshless collocation method by using barycentric Lagrange interpolation basis function based on the Chebyshev nodes to deduce the scheme for solving the three-dimensional Helmholtz equation. First, the spatial variables and their partial derivatives are treated by interpolation basis functions, and the collocation method is established for solving second order differential equations. Then the differential matrix is employed to simplify the differential equations which is on a given test node. Finally, numerical experiments show the accuracy and effectiveness of the proposed method.

The numerical experiments show the advantages of the present method, such as less number of collocation nodes needed, shorter calculation time, higher precision, smaller error and higher efficiency. What is more, the numerical solutions agree well with the exact solutions.

Compared with finite element method, finite difference method and other traditional numerical methods based on grid solution, meshless method can reduce or eliminate the dependence on grid and make the numerical implementation more flexible.

The Helmholtz equation has a wide application background in many fields, such as physics, mechanics, engineering and so on.

This meshless method is first time applied for solving the 3D Helmholtz equation. What is more the present work not only gives the relationship of interpolation nodes but also the test nodes.

The purpose of this paper is to present the benefits of using the Lagrangian relaxation (LR) and subgradient methods in scenario studies for wavelength division multiplexing (WDM) network planning. The problem of WDM network planning for a given set of lightpath demands in a mesh topology network is to select lightpath routes and then allocate wavelength channels to the lightpaths. In WDM network planning, a scenario study is to find out the network performance under different lightpath demands and/or different network resource configurations.

A scenario study must solve a series of related static WDM network planning problems. Each static WDM network planning problem is an optimization problem, and may be formulated as an integer linear programming problem, which can be solved by the proposed Lagrangian relaxation and subgradient methods. This paper uses the Lagrange multipliers that are obtained from previous scenarios as initial Lagrange multiplier values for other related scenarios.

This approach dramatically reduces the computation time for related scenarios. For small to medium variations of scenarios, the method reduces the computation time by several folds. The proposed method is the first method that effectively considers the relations between related scenarios, and uses such relations to improve the computation efficiency of scenario studies in WDM network planning.

The method improves the efficiency of a scenario study in WDM network planning. By using it, many “what‐if” type of scenario study questions can be answered quickly.

Unlike other existing methods that treat each scenario individually, this method effectively uses the information of the relation between different scenarios to improve the overall computation efficiency.

The purpose of this paper is to propose a combined reordering scheme with a wide range of application, called Reversed Cuthill-McKee-approximate minimum degree (RCM-AMD), to improve a preconditioned general minimal residual method for solving equations using Lagrange multiplier method, and facilitates the choice of the reordering for the iterative method.

To reordering the coefficient matrix before a preconditioned iterative method will greatly impact its convergence behavior, but the effect is very problem-dependent, even performs very differently when different preconditionings applied for an identical problem or the scale of the problem varies. The proposed reordering scheme is designed based on the features of two popular ordering schemes, RCM and AMD, and benefits from each of them.

Via numerical experiments for the cases of various scales and difficulties, the effects of RCM-AMD on the preconditioner and the convergence are investigated and the comparisons of RCM, AMD and RCM-AMD are presented. The results show that the proposed reordering scheme RCM-AMD is appropriate for large-scale and difficult problems and can be used more generally and conveniently. The reason of the reordering effects is further analyzed as well.

The proposed RCM-AMD reordering scheme preferable for solving equations using Lagrange multiplier method, especially considering that the large-scale and difficult problems are very common in practical application. This combined reordering scheme is more wide-ranging and facilitates the choice of the reordering for the iterative method, and the proposed iterative method has good performance for practical cases in in-house and commercial codes on PC.

A three-dimensional (3D) unsteady potential flow might admit a variational principle. The purpose of this paper is to adopt a semi-inverse method to search for the variational formulation from the governing equations.

A suitable trial functional with a possible unknown function is constructed, and the identification of the unknown function is given in detail. The Lagrange multiplier method is used to establish a generalized variational principle, but in vain.

Some new variational principles are obtained, and the semi-inverse method can easily overcome the Lagrange crisis.

The semi-inverse method sheds a promising light on variational theory, and it can replace the Lagrange multiplier method for the establishment of a generalized variational principle. It can be used for the establishment of a variational principle for fractal and fractional calculus.

This paper establishes some new variational principles for the 3D unsteady flow and suggests an effective method to eliminate the Lagrange crisis.

An alternative stabilization approach has been developed for the 9‐node Lagrange plane and plate elements. In this approach, a stabilization stiffness is formulated using functions associated with the spurious zero‐energy modes. Efficiency has been increased by employing the same uniformly‐reduced integration scheme on the stabilization and underintegrated stiffness matrices. The results obtained using this rank‐sufficient element, termed the γ‐ψ element, appear to surpass those obtained with other rank‐sufficient 9‐node elements in accuracy.

This study considers a Lagrange direct way of modal assurance criterion (MAC) values of an undamped system. The mentioned method for the sensitivity analysis of the MAC of a sliding machine working table is more close to the exact solution and time efficient. The paper aims to discuss these issues.

Using the Lagrange multipliers to compute the first and second-order sensitivity analysis of MAC values.

Because of the Lagrange multiplier without considering the number of design parameters, one only needs to perform the calculation once. Compared with the indirect way, the direct way is more effective when the number of design parameters is greater than one. This calculation procedure is simple and accurate, which can be popularized and used.

Engineering structure often requires only some structure design, and most of the sub-structure design variables are not related to each other. In this case, this way is better and more efficient. The direct way can be applied to the dynamic optimization design of large structures, the frequency and the mode sensitivity analysis in the process of model modification.

The purpose of this paper is to propose an efficient iterative method for large-scale finite element equations of bad numerical stability arising from deformation analysis with multi-point constraint using Lagrange multiplier method.

In this paper, taking warpage analysis of polymer injection molding based on surface model as an example, the performance of several popular Krylov subspace methods, including conjugate gradient, BiCGSTAB and generalized minimal residual (GMRES), with diffident Incomplete LU (ILU)-type preconditions is investigated and compared. For controlling memory usage, GMRES(m) is also considered. And the ordering technique, commonly used in the direct method, is introduced into the presented iterative method to improve the preconditioner.

It is found that the proposed preconditioned GMRES method is robust and effective for solving problems considered in this paper, and approximate minimum degree (AMD) ordering is most beneficial for the reduction of fill-ins in the ILU preconditioner and acceleration of the convergence, especially for relatively accurate ILU-type preconditioning. And because of concerns about memory usage, GMRES(m) is a good choice if necessary.

In this paper, for overcoming difficulties of bad numerical stability resulting from Lagrange multiplier method, together with increasing scale of problems in engineering applications and limited hardware conditions of computer, a stable and efficient preconditioned iterative method is proposed for practical purpose. Before the preconditioning, AMD reordering, commonly used in the direct method, is introduced to improve the preconditioner. The numerical experiments show the good performance of the proposed iterative method for practical cases, which is implemented in in-house and commercial codes on PC.

This paper aims to carry out assembly variation analysis for mechanisms with compliant joints by considering deformations induced by manufactured deviations. Such an analysis procedure extends the application area of direct linearization method (DLM) to compliant mechanisms and also illustrates the dimensional interaction within multi-loop compliant structures.

By applying DLM to both geometrical equations and Lagrange’s equations of the second kind, an analytical deviation modeling method for mechanisms with compliant joints are proposed and further used for statistical assembly variation analysis. The precision of this method is verified by comparing it with finite element simulation and traditional DLM.

A new modeling method is proposed to represent kinematic relationships between joint deformations and parts/components deviations. Based on a case evaluation, the computational efficiency is improved greatly while the modeling accuracy is maintained at more than 94% rate comparing with the benchmark finite element simulation.

The Equilibrium Equations of Incremental Forces derived from Lagrange’s equations are proposed to quantitatively represent the relationships between manufactured deviations and assembly deformations. The present method extends the application area of DLM to compliant structures, such as automobile suspension systems and some Micro-Electro-Mechanical-Systems.