Advantages Of Finite Difference Methods Research Articles

Semi-analytical sensitivity analysis approach for fully coupled optimization of flexible multibody systems

Flexible multibody systems have high geometric nonlinearity, which makes it very difficult to solve the dynamic response optimization problem. If mathematical programming tools are used, two main optimization methods are weakly coupled methods and fully coupled methods. The latter must consider the full dynamics of the system. When the objective function and constraint function are related to velocity vector or multicomponent-based constraints are involved in the optimization problem, only fully coupled optimization methods can be used. In this case, dynamic sensitivity analysis is necessary, but the solution is relatively difficult. In this paper, a semi-analytical sensitivity analysis approach for fully coupled optimization of flexible multibody systems is proposed. This approach combines the advantages of finite difference method and analytic method, and solves the bottleneck problem of the sensitivity calculation for fully coupled optimization methods. The formulas of the semi-analytical sensitivity analysis approach at the system and element levels are given. And the correctness and effectiveness are verified by two numerical examples. The results show that fully coupled optimization of flexible multibody system dynamics has good engineering application value.

Modular finite volume approach for 3D magnetotelluric modeling of the Earth medium with general anisotropy

Electrical property of the Earth should be best depicted by anisotropic 3D models, while many numerical issues remain to be solved before it becomes practical, such as implementation complexity and computational efficiency. Magnetotellurics (MT) is the primary method to explore electrical property of the deep Earth, and MT forward modeling with 3D anisotropy has been investigated primarily by finite element methods (FEM) and finite difference methods (FDM). Here, we present a new implementation of finite volume method (FVM) accounting for general anisotropy. Theoretically, FVM can combine the advantages of FEM in mesh flexibility and superior accuracy, and the advantages of FDM in straightforward implementation and economic computation load. For demonstration, a modular framework to approximate Maxwell's equations on a structured staggered-grid (mesh flexibility ignored), which solves electric fields tangential to cell edges, is implemented in an object-oriented paradigm. Synthetic examples are presented to verify its accuracy and feasibility, which includes (quasi-) analytic solution of simplified models and numerical solution of a dyke model. A variant of COMMEMI 3D-2 model with general anisotropy is simulated to probe the characteristic behaviors of EM fields under such circumstance. This approach makes the dependence of model operators on anisotropic conductivity parameters explicit and can simplify further sensitivity derivation, and takes a step forward for the feasible development of MT inversion algorithms concerning 3D anisotropy.

An improved method for the calculation of unsaturated\u2013saturated water flow by coupling the FEM and FDM

Numerical modeling of water movement in both unsaturated soils and saturated groundwater aquifers is important for water resource management simulations. The development of efficient numerical algorithms for coupling unsaturated and saturated flow has been a long-lasting challenge in hydrologic modeling, especially for regional-scale modeling. In this study, a new method coupling the Finite Element Method (FEM) and Finite Difference Method (FDM), FE-FDM, is developed to solve Richards equation for simulating unsaturated–saturated water flow. The FEM is adopted to discretize the governing equation in the horizontal direction, while the FDM is used in the vertical direction. This method combines the advantages of FEM in domain discretization, especially for complex computational domain, and the advantages of FDM in modeling simplicity and efficiency. The validity of the new method is demonstrated with three test cases that show that the FE-FDM solutions are accurate and are applicable for regional scale problems. In the test cases, the FE-FDM solutions are compared with experimental data and numerical results calculated with common software packages including FEFLOW and COMSOL. This study verified that the FE-FDM is applicable for simulating water flow in the unsaturated–saturated zone.

Ab-Initio Molecular Dynamics of Organic Compounds on a Massively Parallel Computer

AbstractWe present results of ab-initio electronic structure calculations and molecular dynamics simulations of organic molecules carried out using adaptive curvilinear coordinates, within the local density approximation of density functional theory. This approach allows for an accurate treatment of first-row elements, which makes it particularly suitable for investigations of organic compounds. A recent formulation of this method relies on a real-space approach which combines the advantages of finite-difference methods with the accuracy of adaptive coordinates, and is well suited for implementation on massively parallel computers. We used molecular dynamics simulations to obtain the fully relaxed structures of nitrosyl fluoride (FNO), and of the aromatic heterocycles furan and pyrrole. The equilibrium geometries obtained show excellent agreement with experimental data. The harmonic vibrational frequencies of furan and pyrrole were calculated by diagonalization of their dynamical matrix and are found to agree with experimental data within an rms error of 25 cm-1 and 28 cm-1 for furan and pyrrole respectively. This accuracy is comparable to that attained for smaller organic molecules using elaborate quantum chemistry methods.

An explicit-implicit water-level model for tidal computations of rivers

The combined use of explicit and implicit time integration methods leads to a numerical model, based on the finite element method, that overcomes the often claimed advantages of finite difference methods. The implicit part of the model shows the whole range of flexibility the finite element method is famous for, whereas the explicit part reduces to simple difference quotients for regular and irregular discretizations. The stability of the numerical solution is ensured and high numerical damping is avoided by means of a generalized collocation method for the integrations in space.