Strong-form framework for solving boundary value problems with geometric nonlinearity

We propose an incremental-iterative algorithm by using the strong form collocation method for solving geometric nonlinear problems. As nonlinear analyses concerning large deformation have been relied on the weak form-based methods such as the finite element methods and the reproducing kernel particle methods, the recently developed strong form collocation methods could be new research directions in that the mesh control and quadrature rule are abandoned in the collocation methods. In this work, the radial basis collocation method is adopted to perform the nonlinear analysis. The corresponding parameters affecting the deformation paths such as the increment of applied traction and shape parameter of the radial basis function are discussed. We also investigate the possibility of using the weighted collocation methods in the nonlinear analysis.

Evaluation of meshless radial basis collocation method (RBCM) for heterogeneous conduction and simulation of temperature inside the biological tissues

A weighted meshfree collocation method for incompressible flows using radial basis functions

The direct approximation of strong form using radial basis functions (RBFs), commonly called the radial basis collocation method (RBCM), has been recognized as an effective means for solving boundary value problems. Nevertheless, the non‐compactness of the RBFs precludes its application to problems with local features, such as fracture problems, among others. This work attempts to apply RBCM to fracture mechanics by introducing a domain decomposition technique with proper interface conditions. The proposed method allows (1) natural representation of discontinuity across the crack surfaces and (2) enrichment of crack‐tip solution in a local subdomain. With the proper domain decomposition and interface conditions, exponential convergence rate can be achieved while keeping the discrete system well‐conditioned. The analytical prediction and numerical results demonstrate that an optimal dimension of the near‐tip subdomain exists. The effectiveness of the proposed method is justified by the mathematical analysis and demonstrated by the numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.

This paper briefly describes the development of generalized collocation methods for solving coupled systems of nonlinear, one-dimensional, parabolic, partial differential equations. parabolic, partial differential equations. The methods are applied to several problems that are representative of fluid flow in a porous medium. The numerical results for the problems solved indicate that collocation methods using C piecewise polynomials with Gauss-Legendre collocation points polynomials with Gauss-Legendre collocation points are more efficient than conventional, second-order, finite-difference methods. In particular, for problems whose solutions are smooth, one obtains more accuracy per unit of computer time with increasingly higher-order collocation methods. For problems whose solutions are characterized by a steep, shock-like, transient wave front, one can conclude that increasing the order of the collocation method does not necessarily result in a more efficient method, but high-accuracy collocation methods still appear to be more efficient than conventional, second-order, finite-difference methods. Also, for a given accuracy, collocation methods generally require less computer storage than conventional, second-order, finite-difference methods. Finally, the collocation technique presented here is shown to be not directly applicable to the saturation equation for immiscible flow. Introduction The purpose of this paper is to investigate collocation methods for the numerical solution of nonlinear equations similar to those that describe the flow of a fully compressible fluid in a porous medium and those that describe the process by which one miscible liquid displaces another liquid in a porous medium. The objective is to determine if collocation methods are applicable and economically justified for solving these types of petroleum engineering problems. The conclusions petroleum engineering problems. The conclusions reached are based on the amount of computer time and computer storage that is required to obtain a specified accuracy. Since only two problems are solved, the results may not be extendable to more general problems; however, experience in solving a wide variety of problems indicates that this is the general behavior that might be expected when using collocation methods. Previously published results indicate how Galerkin, collocation, and finite-element methods can be used to solve various problems that arise in petroleum engineering. Most of these papers present a variety of arguments in favor of these present a variety of arguments in favor of these methods. The principal argument is that better answers can be obtained for the same computational effort than by finite-difference methods. Culham and Varga have presented the most convincing evidence in favor of Galerkin methods. However, they only considered piecewise linear and piecewise cubic-basis functions and they only considered one problem. Because of these limitations, it is not problem. Because of these limitations, it is not clear whether their conclusions can be extended to higher-order Galerkin methods or to other types of problems. Also, they did not consider collocation problems. Also, they did not consider collocation methods in any detail. This paper considers the use of increasingly higher-order collocation methods and compares the results with those obtained with the usual second-order, finite-difference method. The Galerkin method is not considered, even though that is the method with which collocation must ultimately compete. This was done for several reasons. A preliminary operation count indicates that Galerkin methods will not be competitive with collocation methods for most problems. Also, it is not clear how to implement the higher-order Galerkin methods to achieve maximum efficiency. These remarks indicate that the relative merits of collocation and Galerkin methods are still uncertain. PROBLEMS SOLVED PROBLEMS SOLVED Two problems are solved to serve as a basis for the comparisons. The first problem, called the "gas flow problem," is the same problem considered by Culham and Varga. This problem is nonlinear and includes a volumetric source term. SPEJ P. 345

Isogeometric collocation method based on residual parameterization of planar physical domain

This article examines the unique solvability of a boundary value problem for hyperbolic systems under periodic conditions. To solve the stated problem, the Dzhumabaev method is employed, which transforms the original boundary value problem into an equivalent multipoint boundary value problem for ordinary differential equations with parameters, incorporating both initial and additional boundary conditions. Furthermore, the study establishes a connection between the unique solvability of the original problem and the invertibility of the matrix . Recurrent relations for determining the elements of the inverse matrix are also derived.

On solvability and unique solvability of two-point singular boundary value problems

Sufficient conditions are established for the solvability of the boundary value problem where p : C(I; Rn ) × C(I; Rn ) → L(I; Rn ), q : C(I; Rn ) → L(I; Rn ), l : C(I; Rn ) × C(I; Rn ) → Rn , and cn : C(I; Rn ) → Rn are continuous operators, and p(x, ·) and l(x, ·) are linear operators for any fixed x ∈ C(I; Rn ).

Challenges in motion planning for multiple quadrotors in complex environments lie in overall flight efficiency and the avoidance of obstacles, deadlock, and collisions among themselves. In this paper, we present a gradient-free trajectory generation method for multiple quadrotors in dynamic obstacle-dense environments with the consideration of time consumption. A model predictive control (MPC)-based approach for each quadrotor is proposed to achieve distributed and asynchronous cooperative motion planning. First, the motion primitives of each quadrotor are formulated as the boundary state constrained primitives (BSCPs) which are constructed with jerk limited trajectory (JLT) generation method, a boundary value problem (BVP) solver, to obtain time-optimal trajectories. They are then approximated with a neural network (NN), pre-trained using this solver to reduce the computational burden. The NN is used for fast evaluation with the guidance of a navigation function during optimization to guarantee flight safety without deadlock. Finally, the reference trajectories are generated using the same BVP solver. Our simulation and experimental results demonstrate the superior performance of the proposed method.

Purpose – A variety of meshless methods have been developed in the last 20 years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The purpose of this paper is to develop an efficient and accurate algorithms based on meshless methods for the solution of problems involving both material and geometrical nonlinearities. Design/methodology/approach – A parallel two-dimensional linear elastic computer code is presented for a maximum entropy basis functions based meshless method. The two-dimensional algorithm is subsequently extended to three-dimensional adaptive nonlinear and three-dimensional parallel nonlinear adaptively coupled finite element, meshless method cases. The Prandtl-Reuss constitutive model is used to model elasto-plasticity and total Lagrangian formulations are used to model finite deformation. Furthermore, Zienkiewicz and Zhu and Chung and Belytschko error estimation procedure are used in the FE and meshless regions of the problem domain, respectively. The message passing interface library and open-source software packages, METIS and MUltifrontal Massively Parallel Solver are used for the high performance computation. Findings – Numerical examples are given to demonstrate the correct implementation and performance of the parallel algorithms. The agreement between the numerical and analytical results in the case of linear elastic example is excellent. For the nonlinear problems load-displacement curve are compared with the reference FEM and found in a very good agreement. As compared to the FEM, no volumetric locking was observed in the case of meshless method. Furthermore, it is shown that increasing the number of processors up to a given number improve the performance of parallel algorithms in term of simulation time, speedup and efficiency. Originality/value – Problems involving both material and geometrical nonlinearities are of practical importance in many engineering applications, e.g. geomechanics, metal forming and biomechanics. A family of parallel algorithms has been developed in this paper for these problems using adaptively coupled finite element, meshless method (based on maximum entropy basis functions) for distributed memory computer architectures.

Second-order inelastic analysis of cable-stayed bridges

A multi-material ALE model for investigating impact dynamics of pile-soil systems

Meshless methods have some advantages over their counterparts such as the finite-element method (FEM). However, existing meshless methods for computational electromagnetic fields are still not as efficient as FEM. In this paper, we compare two meshless methods of discretizing the computational domain of Poisson-like problems; namely, the point collocation and Galerkin methods (which use the strong and weak forms of the governing equation respectively), and their effects on the computational accuracy and efficiency of the magnetic fields. We also discuss methods of handling discontinuities at the material interface. We present several examples, which also provide a means to validate and evaluate both meshless methods. Exact solutions and/or FEM are used as a basis for comparison. In addition, we also verify the results by comparing computed magnetic forces against those measured experimentally

Nonlinear aerostatic stability analysis of suspension bridges