Application of mixed meshless solution procedures for deformation modeling in gradient elasticity

The present study is related to the utilization of the mixed Meshless Local PetrovGalerkin (MLPG) methods for solving problems in gradient elasticity, which are governed by fourth-order differential equations. Here, three different numerical MLPG methods are presented, where the continuity requirements for the approximation functions are lowered by applying different mixed procedures to improve the numerical accuracy and efficiency. The first one is based on the direct solution of the problem, where the primary variable (displacement) and its independently chosen higher-order variables are approximated separately. The global discretized system of equations consists of appropriate equilibrium and compatibility equations written for each node and the solution vector contains all unknown independent nodal variables. Such approach demands only the first-order continuity of meshless approximation functions. The second and third procedures are both based on the displacement-based operator-split approach, where the original gradient elasticity problem is solved as two uncoupled problems governed by the second-order differential equations. Herein, in both uncoupled problems only primary variable (displacement) and its first derivative (strain) are approximated independently. In these procedures the original problem is solved by a staggered approach, where the solution of the first uncoupled equation is utilized as an input in the second equation. The main difference in the second and third procedure is that the one is based on the solution of the local weak forms of the governing equations, while the other is based on solution of the strong forms of the same equations. The accuracy of the presented computational methods is compared to analytical solutions and demonstrated on a one-dimensional benchmark problem of axial bar in gradient elasticity

A Meshfree Generalized Finite Difference Method for Surface PDEs

In this paper, we propose a novel meshfree Generalized Finite Difference Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative approximations for the same are done directly on the tangent space, in a manner that mimics the procedure followed in volume-based meshfree GFDMs. As a result, the proposed method not only does not require a mesh, it also does not require an explicit reconstruction of the manifold. In contrast to existing methods, it avoids the complexities of dealing with a manifold metric, while also avoiding the need to solve a PDE in the embedding space. A major advantage of this method is that all developments in usual volume-based numerical methods can be directly ported over to surfaces using this framework. We propose discretizations of the surface gradient operator, the surface Laplacian and surface Diffusion operators. Possibilities to deal with anisotropic and discontinous surface properties (with large jumps) are also introduced, and a few practical applications are presented

On Meshfree GFDM Solvers for the Incompressible Navier-Stokes Equations

Meshfree solution schemes for the incompressible Navier--Stokes equations are usually based on algorithms commonly used in finite volume methods, such as projection methods, SIMPLE and PISO algorithms. However, drawbacks of these algorithms that are specific to meshfree methods have often been overlooked. In this paper, we study the drawbacks of conventionally used meshfree Generalized Finite Difference Method~(GFDM) schemes for Lagrangian incompressible Navier-Stokes equations, both operator splitting schemes and monolithic schemes. The major drawback of most of these schemes is inaccurate local approximations to the mass conservation condition. Further, we propose a new modification of a commonly used monolithic scheme that overcomes these problems and shows a better approximation for the velocity divergence condition. We then perform a numerical comparison which shows the new monolithic scheme to be more accurate than existing schemes

Meshless 2D direct numerical simulation and heat transfer in a backward-facing step with heat conduction in the step

A meshless direct pressure-velocity coupling procedure is presented to perform Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES) of turbulent incompressible flows in regular and irregular geometries. The proposed method is a combination of several efficient techniques found in different Computational Fluid Dynamic (CFD) procedures. With this new procedure, preliminary calculations with 2D steady state flows show that viscous effects become negligible faster that ever predicted numerically. The fundamental idea of this method lays on several important inconsistencies found in three of the most popular techniques used in CFD, segregated procedures, as well as in other formulations. The inconsistencies found become important in elliptic flows and they might lead to some wrong solutions. Preliminary calculations done in 2D laminar flows, suggest that the numerical diffusion and interpolation error are much important at low speeds, mainly when both, viscous and inertia forces are present. With this competitive and efficient procedure, the solution of the 2D Direct Numerical Simulation of turbulent flow with heat transfer on a backward-facing step is presented. The thermal energy is going to be transferred to the fluid through conduction on the step, with both constant temperature and heat flux conditions in the back wall of the step. The variation of the local Nusselt Number through the wall will be studied and its corresponding effect in the energy transfer to the fluid

Two dimensional solution of the advection-diffusion equation using two collocation methods with local upwinding RBF

The two-dimensional advection-diffusion equation is solved using two local collocation methods with Multiquadric (MQ)Radial Basis Functions (RBFs). Although both methods use upwinding, the first one, similar to the method of Kansa, approximates the dependent variable with a linear combination of MQs. The nodes are grouped into two types of stencil: cross-shaped stencil to approximate the Laplacian of the variable and circular sector shape stencil to approximate the gradient components. The circular sector opens in opposite to the flow direction and therefore the maximum number of nodes and the shape parameter value are selected conveniently. The second method is based on the Hermitian interpolation where the approximation function is a linear combination of MQs and the resulting functions of applying partial differential equation (PDE) and boundary operators to MQs, all of them centred at different points. The performance of these methods is analysed by solving several test problems whose analytical solutions are known. Solutions are obtained for different Peclet numbers, Pe, and several values of the shape parameter. For high Peclet numbers the accuracy of the second method is affected by the ill-conditioning of the interpolation matrix while the first interpolation method requires the introduction of additional nodes in the cross stencil. For low Pe both methods yield accurate results. Moreover, the first method is employed to solve the twodimensional Navier-Stokes equations in velocity-vorticity formulation for the lid-driven cavity problem moderate Pe

Direct solution of Navier-Stokes equations by using an upwind local RBF-DQ method

The differential quadrature (DQ) method is able to obtain quite accurate numerical solutions of differential equations with few grid points and less computational effort. However, the traditional DQ method is convenient only for regular regions and lacks upwind mechanism to characterize the convection of the fluid flow. In this paper, an upwind local radial basis function-based DQ (RBF-DQ) method is applied to solve the Navier-Stokes equations, instead of using an iterative algorithm for the primitive variables. The non-linear collocated equations are solved using the Levenberg-Marquardt method. The irregular regions of 2D channel flow with different obstructions situations are considered. Finally, the approach is validated by comparing the results with those obtained using the well-validated Fluent commercial package