Strong-and weak-Form meshless methods in computational biomechanics

Meshless methods (MMs) were introduced in the late 1970s to solve problems in astrophysics. In MMs the spatial domain is represented by a set of nodes (cloud of points) and not discretized by elements as in most of the mesh-based methods (finite difference method, finite element method, finite volume method); consequently, there is no need for predefined connectivity between the nodes. In this chapter we are going to give an overview of applications, advantages, and disadvantages of various MMs developed and applied in the context of computational biomechanics. Strong and weak formulations will be presented, focusing on the novel interpolation schemes such as modified moving least squares and discretization correction particle strength exchange method, along with the meshless total Lagrangian explicit dynamics method. The applicability of the methods in multiscale problems and their inherent parallelization will be depicted through various applications, along with their advantages over the traditional mesh-based numerical methods. MMs can be considered as mainstream numerical methods able to tackle demanding engineering applications. Intensive and rigorous research in the field will make MMs robust enough to be used by industry

Strong-and weak-Form meshless methods in computational biomechanics

Meshless methods (MMs) were introduced in the late 1970s to solve problems in astrophysics. In MMs the spatial domain is represented by a set of nodes (cloud of points) and not discretized by elements as in most of the mesh-based methods (finite difference method, finite element method, finite volume method); consequently, there is no need for predefined connectivity between the nodes. In this chapter we are going to give an overview of applications, advantages, and disadvantages of various MMs developed and applied in the context of computational biomechanics. Strong and weak formulations will be presented, focusing on the novel interpolation schemes such as modified moving least squares and discretization correction particle strength exchange method, along with the meshless total Lagrangian explicit dynamics method. The applicability of the methods in multiscale problems and their inherent parallelization will be depicted through various applications, along with their advantages over the traditional mesh-based numerical methods. MMs can be considered as mainstream numerical methods able to tackle demanding engineering applications. Intensive and rigorous research in the field will make MMs robust enough to be used by industry

An explicit meshless point collocation method for electrically driven magnetohydrodynamics (MHD) flow

In this paper, we develop a meshless collocation scheme for the numerical solution of magnetohydrodynamics (MHD) flow equations. We consider the transient laminar flow of an incompressible, viscous and electrically conducting fluid in a rectangular duct. The flow is driven by the current produced by electrodes placed on the walls of the duct. The method combines a meshless collocation scheme with the newly developed Discretization Corrected Particle Strength Exchange (DC PSE) interpolation method. To highlight the applicability of the method, we discretize the spatial domain by using uniformly (Cartesian) and irregularly distributed nodes. The proposed solution method can handle high Hartmann (Ha) numbers and captures the boundary layers formed in such cases, without the presence of unwanted oscillations, by employing a local mesh refinement procedure close to the boundaries. The use of local refinement reduces the computational cost. We apply an explicit time integration scheme and we compute the critical time step that ensures stability through the Gershgorin theorem. Finally, we present numerical results obtained using different orientation of the applied magnetic field

An implicit potential method along with a meshless technique for incompressible fluid flows for regular and irregular geometries in 2D and 3D

We present the Implicit Potential (IPOT) numerical scheme developed in the framework of meshless point collocation. The proposed scheme is used for the numerical solution of the steady state, incompressible Navier-Stokes (N-S) equations in their primitive variable (u-v-w-p) formulation. The governing equations are solved in their strong form using either a collocated or a semi-staggered type meshless nodal configuration. The unknown field functions and derivatives are calculated using the Modified Moving Least Squares (MMLS) interpolation method. Both velocity-correction and pressure-correction methods applied ensure the incompressibility constraint and mass conservation. The proposed meshless point collocation (MPC) scheme has the following characteristics: (i) it can be applied, in a straightforward manner to: steady, unsteady, internal and external fluid flows in 2D and 3D, (ii) it equally applies to regular an irregular geometries, (iii) a distribution of points is sufficient, no numerical integration in space nor any mesh structure are required, (iv) there is no need for pressure boundary conditions since no pressure constitutive equation is solved, (v) it is quite simple and accurate, (vi) results can be obtained using collocated or semi-staggered nodal distributions, (vii) there is no need to compute the velocity potential nor the unit normal vectors and (viii) there is no need for a curvilinear system of coordinates. Simulations of fluid flow in 2D and 3D for regular and irregular geometries indicate the validity of the proposed methodology

Strong-form approach to elasticity: Hybrid finite difference-meshless collocation method (FDMCM)

We propose a numerical method that combines the finite difference (FD) and strong form (collocation) meshless method (MM) for solving linear elasticity equations. We call this new method FDMCM. The FDMCM scheme uses a uniform Cartesian grid embedded in complex geometries and applies both methods to calculate spatial derivatives. The spatial domain is represented by a set of nodes categorized as (i) boundary and near boundary nodes, and (ii) interior nodes. For boundary and near boundary nodes, where the finite difference stencil cannot be defined, the Discretization Corrected Particle Strength Exchange (DC PSE) scheme is used for derivative evaluation, while for interior nodes standard second order finite differences are used. FDMCM method combines the advantages of both FD and DC PSE methods. It supports a fast and simple generation of grids and provides convergence rates comparable to weak formulations. We demonstrate the appropriateness and robustness of the proposed scheme through various benchmark problems in 2D and 3D. Numerical results show good accuracy and h-convergence properties. The ease of computational grid generation makes the method particularly suited for problems where geometries are very complicated and known only imperfectly from images, frequently occurring in e.g. geomechanics and patient-specific biomechanics, where the proposed FDMCM method, after its extension to non-linear regime, appears to be a promising alternative to the traditional weak form-based numerical schemes used in the field