Hybrid mesh/particle meshless method for modeling geological flows with discontinuous transport properties

In the present paper, we introduce the Finite Difference Method-Meshless Method (FDM-MM) in the context of geodynamical simulations. The proposed numerical scheme relies on the well-established FD method along with the newly developed “meshless” method and, is considered as a hybrid Eulerian/Lagrangian scheme. Mass, momentum, and energy equations are solved using an FDM method, while material properties are distributed over a set of markers (particles), which represent the spatial domain, with the solution interpolated back to the Eulerian grid. The proposed scheme is capable of solving flow equations (Stokes flow) in uniform geometries with particles, “sprinkled” in the spatial domain and is used to solve convection- diffusion problems avoiding the oscillation produced in the Eulerian approach. The resulting algebraic linear systems were solved using direct solvers. Our hybrid approach can capture sharp variations of stresses and thermal gradients in problems with a strongly variable viscosity and thermal conductivity as demonstrated through various benchmarking test cases. The present hybrid approach allows for the accurate calculation of fine thermal structures, offering local type adaptivity through the flexibility of the particle method

Επίλυση προβλημάτων υπολογιστικής ρευστομηχανικής σε αιμοφόρα αγγεία με ταύτιση λύσεων σε κατανεμημένα σημεία (κόμβους) στο πεδίο ροής του με τη μέθοδο της μη πλεγματικής διαμόρφωσης (meshless method)

The aim of present doctoral thesis is double fold and, includes research activity both in the fields of applied and basic research. More precisely, it includes the application of modern numerical methods (Finite Element Method, Finite Volume Method) for the study of blood flow, as well as the development of modern numerical methodologies, which do not rely on the use of a computational mesh, that is the socalled meshless or Meshfree methods. Furthermore, the effective application of sophisticated numerical methods in the medical practice (Simulation Treatment Planning) has been studied, since there is a great necessity for effective prevention, diagnosis and therapeutic confrontation of illnesses the cardiac and vascular system. The simulation conducted they aim to assist the doctor in the decision-making. At the same time, regarding of area of the basic research, sophisticated numerical methods were developed and applied to various applications of science and engineering. More precisely, results will be presented for Computational Fluid Dynamics problems.O σκοπός της παρούσας διδακτορικής διατριβής είναι διττός και, εμπεριέχει δραστηριότητα τόσο στο κομμάτι της εφαρμοσμένης όσο και της βασικής έρευνας. Πιο συγκεκριμένα, περιλαμβάνει την εφαρμογή σύγχρονων υπολογιστικών μεθόδων (Μέθοδος Πεπερασμένων Στοιχείων, Μέθοδος Πεπερασμένων Όγκων) στη μελέτη της ροής του αίματος, καθώς και την ανάπτυξη σύγχρονων υπολογιστικών μεθοδολογιών που δε στηρίζονται στη χρήση πλέγματος. Ταυτόχρονα μελετάται η αποτελεσματική εφαρμογή των μεθόδων της Υπολογιστικής Ρευστομηχανικής στην ιατρική πρακτική (Simulation Treatment Planning), για την αποτελεσματική πρόληψη, διάγνωση και θεραπευτική αντιμετώπιση των νόσων του καρδιακού και περιφερικού αγγειακού συστήματος. Η διαδικασία υλοποίησης των προσομοιώσεων έχουν σκοπό την υποβοήθηση του θεράποντα ιατρού στη λήψη ιατρικής απόφασης σχετικά με τη θεραπευτική αγωγή. Παράλληλα, στο κομμάτι της βασικής έρευνας αναπτύσσονται σύγχρονες υπολογιστικές μέθοδοι οι οποίες πρόκειται να καλύψουν τις αδυναμίες που παρουσιάζουν οι διαδεδομένες υπολογιστικές μέθοδοι. Η υλοποίησή τους βρίσκει ανταπόκριση και πεδίο εφαρμογής σε διάφορους τομείς της Επιστήμης και της Μηχανικής. Έτσι, θα παρουσιαστούν αποτελέσματα μόνο στο τομέα της Υπολογιστικής Ρευστομηχανικής. Τα κριτήρια αυτά θα στηριχθούν σε ποσοτικές συνδυαστικές αναλύσεις, οι οποίες ενσωματώνουν το state-of-the-art της μορφολογικής απεικόνισης των αγγείων στο state-of-the-art των μεθοδολογιών της υπολογιστικής ρευστομηχανικής, οι οποίες είναι άμεσα σχετιζόμενες με την εκτίμηση των αιμοδυναμικών παραγόντων και τάσεων που εφαρμόζονται σε πάσχουσες περιοχές του αγγειακού συστήματος, όπως στενώσεις, θρόμβοι και ανευρύσματα

MHD natural-convection flow in an inclined square enclosure filled with a micropolar-nanofluid

Transient, laminar, natural-convection flow of a micropolar-nanofluid (Al2O3/water) in the presence of a magnetic field in an inclined rectangular enclosure is considered. A meshless point collocation method utilizing a velocity-correction scheme has been developed. The governing equations in their velocity–vorticity formulation are solved numerically for various Rayleigh (Ra) and Hartman (Ha) numbers, different nanoparticles volume fractions (φ) and considering different inclination angles and magnetic field directions. The results show that, both, the strength and orientation of the magnetic field significantly affect the flow and temperature fields. For the cases considering herein, experimentally given forms of dynamic viscosity, thermal conductivity and electrical conductivity are utilized

Solution of Two-dimensional Linear and Nonlinear Unsteady Schrödinger Equation using “Quantum Hydrodynamics” Formulation with a MLPG Collocation Method

A numerical solution of the linear and nonlinear time-dependent Schrödinger equation is obtained, using the strong form MLPG Collocation method. Schrödinger equation is replaced by a system of coupled partial differential equa tions in terms of particle density and velocity potential, by separating the real and imaginary parts of a general solution, called a quantum hydrodynamic (QHD) equa tion, which is formally analogous to the equations of irrotational motion in a classical fluid. The approximation of the field variables is obtained with the Moving Least Squares (MLS) approximation and the implicit Crank-Nicolson scheme is used for time discretization. For the two-dimensional nonlinear Schrödinger equation, the lagging of coefficients method has been utilized to eliminate the non-linearity of the corresponding examined problem. A Type-I nodal distribution is used in order to provide convergence for the discrete Laplacian operator used at the governing equation. Numerical results are validated, comparing them with analyti cal and numerical solutions

An accurate, stable and efficient domain-type meshless method for the solution of MHD flow problems

The aim of the present paper is the development of an efficient numerical algorithm for the solution of magnetohydrodynamics flow problems for regular and irregular geometries subject to Dirichlet, Neumann and Robin boundary conditions. Toward this, the meshless point collocation method (MPCM) is used for MHD flow problems in channels with fully insulating or partially insulating and partially conducting walls, having rectangular, circu- lar, elliptical or even arbitrary cross sections. MPC is a truly meshless and computationally efficient method. The maximum principle for the discrete harmonic operator in the mesh- free point collocation method has been proven very recently, and the convergence proof for the numerical solution of the Poisson problem with Dirichlet boundary conditions have been attained also. Additionally, in the present work convergence is attained for Neumann and Robin boundary conditions, accordingly. The shape functions are constructed using the Moving Least Squares (MLS) approximation. The refinement procedure with meshless methods is obtained with an easily handled and fully automated manner. We present results for Hartmann number up to 105 . The numerical evidences of the proposed meshless method demonstrate the accuracy of the solutions after comparing with the exact solution and the conventional FEM and BEM, for the Dirichlet, Neumann and Robin boundary con- ditions of interior problems with simple or complex boundaries

Localized meshless point collocation method for time-dependent magnetohydrodynamics flow through pipes under a variety of wall conductivity conditions

In this article a numerical solution of the time dependent, coupled system equations of magnetohydrody- namics (MHD) flow is obtained, using the strong-form local meshless point collocation (LMPC) method. The approxima- tion of the field variables is obtained with the moving least squares (MLS) approximation. Regular and irregular nodal distributions are used. Thus, a numerical solver is developed for the unsteady coupled MHD problems, using the collo- cation formulation, for regular and irregular cross sections, as are the rectangular, triangular and circular. Arbitrary wall conductivity conditions are applied when a uniform mag- netic field is imposed at characteristic directions relative to the flow one. Velocity and induced magnetic field across the section have been evaluated at various time intervals for sev- eral Hartmann numbers (up to 105) and wall conductivities. The numerical results of the strong-form MPC method are compared with those obtained using two weak-form mesh- less methods, that is, the local boundary integral equation (LBIE) meshless method and the meshless local Petrov– Galerkin (MLPG) method, and with the analytical solutions, where they are available. Furthermore, the accuracy of the method is assessed in terms of the error norms L 2 and L ∞ , the number of nodes in the domain of influence and the time step length depicting the convergence rate of the method. Run time results are also presented demonstrating the efficiency and the applicability of the method for real world problems

Adaptive support domain implementation on the Moving Least Squares approximation for Mfree methods applied on elliptic and parabolic PDE problems using strong-form description

The extent of application of meshfree methods based on point collocation (PC) techniques with adaptive support domain for strong form Partial Differential Equations (PDE) is investigated. The basis functions are constructed using the Moving Least Square (MLS) approximation. The weak-form description of PDEs is used in most MLS methods to circumvent problems related to the increased level of resolution necessary near natural (Neumann) boundary conditions (BCs), dislocations, or regions of steep gradients. Alternatively, one can adopt Radial Basis Function (RBF) approximation on the strong-form of PDEs using meshless PC methods, due to the delta function behavior (exact solution on nodes). The present approach is one of the few successful attempts of using MLS approximation [Atluri, Liu, and Han (2006), Han, Liu, Rajendran and Atluri (2006), Atluri and Liu (2006)] instead of RBF approximation for the meshless PC method using strong-form description. To increase the accuracy of the MLS interpolation method and its robustness in problems with natural BCs, a suitable support domain should be chosen in order to ensure an optimized area of coverage for interpolation. To this end, the basis functions are constructed using two different approaches, pertinent to the dimension of the support domain. On one hand, a compact form for the support domain is retained by keeping its radius constant. On the other hand, one can control the number of neighboring nodes as the support domain of each point. The results show that some inaccuracies are present near the boundaries using the first approach, due to the limited number of nodes belonging to the support domain, which results in failed matrix inversion. Instead, the second approach offers capability for fully matrix inversion under many (if not all) circumstances, resulting in basis functions of increased accuracy and robustness. This PC method, applied along with an intelligent adaptive refinement, is demonstrated for elliptic and for parabolic PDEs, related to many flow and mass transfer problems

Meshfree Point Collocation Schemes for 2D Steady State Incompressible Navier-Stokes Equations in Velocity-Vorticity Formulation for High Values of Reynolds Number

A meshfree point collocation method has been developed for the velocity- vorticity formulation of two-dimensional, steady state incompressible Navier-Stokes equations. Particular emphasis was placed on the application of the velocity-correc- tion method, ensuring the continuity equation. The Moving Least Squares (MLS) approximation is employed for the construction of the shape functions, in conjunc- tion with the general framework of the point collocation method. Computations are obtained for regular and irregular nodal distributions, stressing the positivity con- ditions that make the matrix of the system stable and convergent. The accuracy and the stability of the proposed scheme are demonstrated through two representative, well-known, and established benchmark problems. The numerical scheme was also applied to a case with irregular geometry for marginally high Reynolds number

Numerical Solution of Non-Isothermal Fluid Flows Using Local Radial Basis Functions (LRBF) Interpolation and a Velocity-Correction Method

Meshfree point collocation method (MPCM) is developed, solving the velocity-vorticity formulation of Navier-Stokes equations, for two-dimensional, steady state incompressible viscous flow problems in the presence of heat transfer. Particular emphasis is placed on the application of the velocity-correction method, ensuring the continuity equation. The Gaussian Radial Basis Functions (GRBF) interpolation is employed to construct the shape functions in conjunction with the framework of the point collocation method. The cases of forced, natural and mixed convection in a 2D rectangular enclosure are examined. The accuracy and the sta- bility of the proposed scheme are demonstrated through three representative, well known and established benchmark problems. Results are presented for high values of the characteristics non-dimensional numbers of the flow, that is, the Reynolds, the Rayleigh and the Richardson numbe