Meshless based CFD


Towards a unified numerical framework for multi-scale CFD

Numerical modeling became a useful tool for engineering design and analysis. There is a plethora of numerical methods applied on a variety of interesting problems, while extensive research is devoted on finding a multi-scale method that can be applied to micro- up to macro-scale. Mesh-based methods, such as Finite Difference Method (FDM), Finite Element Method (FEM) and Finite Volume Method (FVM) dominate the field since they provide accurate and reproducible results for a wide range of applications. Despite their success these methods fail to model problems where materials can freely move around (as in Computational Fluid Dynamics (CFD)) or where large deformations of the material can occur. The predefined connectivity is difficult to maintain and introduces errors in the numerical procedure. The last two decades meshless or meshfree methods entered the field. A meshless unified numerical framework will be introduced dealing with problems in the CFD field. The applicability of meshless methods in both Eulerian and Lagrangian descriptions will be presented for problems in biomechanics, heat transfer, microfluidics and nanofluidics/ferrofluids. Finally, the applicability and the advantages of meshless methods are depicted and highlighted.


Our aim is to develop a meshless based numerical method for the solution of multi-scale and multi-phase problems arising in engineering and science. These methods .


A robust meshless solver for incompressible Navier-Stokes equations

A robust meshless solver has been developed for the numerical simulation of fluid flow problems.

  • Steady state and transient
  • Incompressible and compressible
  • Newtonian and non-Newtonian
  • laminar and turbulent

flow cases were considered. The meshless scheme developed was able to solve the governing flow equations in their

  • weak and strong form,

using the

  • velocity-pressure (primitive variables)

and the

  • velocity-vorticity formulation.

Multi-physics simulations

Tumor ablation simulation using Dynamic Mode Decomposition

Image-guided thermal ablation has been gaining popularity as a minimally invasive treatment of localized solid tumors. Energy sources for the delivery of thermal energy to coagulate and destroy cancerous lesions may include radiofrequency ablation (RFA), microwaves (MW), high-intensity focused ultrasound (HIFU), and light amplification by stimulated emission of radiation (LASER). Exposure to heat may render cancer cells more sensitive to radiation or even directly attack cancer cells that show reduced sensitivity to radiation. As thermal ablation contributes to the damage of cancer cells, usually with minimal injury to normal tissues, it is a powerful alternative to more conventional treatment modalities, suchas chemotherapy and radiation therapy.

We developed a dynamic mode decomposition (DMD) method in conjunction with the meshless point collocation method has been used in order to numerically solve the transient bioheat transfer equation with temperature dependent properties. The classical Pennes bioheat equation has been extended to incorporate water evaporation and tissue damage during ablation, accounting for temperature-dependent thermal properties of the tissue. This approach treats three-dimensional (3D) heat conduction problems within pathological tissues of locally varying conductivity and locally varying blood perfusion rate, viewing them as continuous domains, and solving the corresponding bioheat differential equation with variable coefficients. With the present scheme, the time needed for running a simulation in a standard PC amounts to just a few seconds for high-resolution images.

Flow in porous media

Natural convection in a cavity filled with porous medium

We developed a numerical scheme that can deal with the flow and the conjugate heat transfer phenomena during the natural convection of a nanofluid in an enclosure filled with a porous medium are investigated, and the pertinent equations, including the Brinkman correction and inertial terms, are solved using highly efficient meshless simulation techniques. A heat source is located at the bottom wall of the enclosure, while the rest of the walls are kept at constant temperature. The meshless point collocation method was used in order to solve the nonlinear coupled partial differential equations governing flow and heat transport in the porous medium. The numerical results portray the recirculation phenomena and the temperature distributions within the cavity and provide predictions of the heat transfer rate at the heated wall as a function of the local Nusselt number. Conclusions on the cooling performance of the porous medium attributed to the nanofluid are also drawn and discussed. A modified expression for the thermal expansion coefficient of the nanofluid is also suggested that is self-consistent in terms of the density and solid volume fraction dependence on temperature and differs from the commonly used, simplified, volume-averaging expression. In terms of numerical aspects, the extension of the applicability of meshless methods to physical systems containing nanofluid saturated porous media is presented here, combined with the testing of the behavior of error-correction and stability algorithms for meshless methods, such as the velocity correction technique in such systems. Although the meshless method is applied to simple geometries in this work for the sake of comparison with previous works, it remains easily applicable to a much wider range of geometries and complex systems.

Bile flow in the liver lobule

The aim of this study is to develop a multi-scale mathematical model of liver structure and function that can make predictions for alterations of clinically-relevant parameters such as blood and bile flow, induced by drugs or disease conditions. The general strategy is to combine experiments and theory using the mouse liver as model system. First, we used a pipeline of state-of-the-art imaging techniques and novel image analysis algorithms to reconstruct the 3D-structure of liver tissue and its dynamics in vivo. By applying high- and super-resolution light microscopy and electron microscopy (EM), and intra-vital imaging we developed quantitative understanding of liver tissue organization at multi-scale, ranging from the subcellular to the lobule levels. Second, we applied intra-vital imaging to quantitatively measure bile flux at different positions from the central to portal veins. The quantitative parameters extracted were then used to generate 1) a tissue structure model based on the 3D organization of basal and apical membranes, etc. and 2) a fluid mechanics model, e.g. of bile flux, based on diameter distribution of bile canaliculi (BC), apical markers of bile secretion activity, graph connectivity and cardinality, and 3) flux estimation from in vivo intra-vital imaging. Predictions from the models were tested by pharmacological perturbations to alter predictably acto-myosin contractility, leading to alterations of the apical surface of hepatocytes and, consequently, of the BC network and bile flux. Validation studies were conducted on the liver by intra-vital imaging. The multi-scale model resulted from the integration of low-level models (i.e. molecular pathways, sub-cellular organelles) with high-level models (cell-cell contacts, fluid mechanics), thus bridging the molecular to the organ scale.


Today geodynamic modeling has established itself as a fundamental branch of the Earth Sciences. On the one hand, direct observations of geodynamic processes are usually limited in both time and space. On the other hand, numerical methods are capable of simulating millions of years of geodynamic processes in a matter of days on a desktop computer. We have developed a 2D thermo-mechanical meshless Eulerian/Lagrangian collocation technique. Mass and momentum equations are solved using a meshless point collocation Eulerian method, while the energy equations are solved using a set of particles, distributed over the spatial domain, with the solution interpolated back to the Eulerian grid at every time step. This hybrid approach allows us to accurately calculate fine thermal structures. The shape functions were constructed using the Discretization Correction Particle Strength Exchange (DC PSE) method. The proposed scheme allows us to solve the flow equations (Stokes flow) in fully irregular geometries while particles, “sprinkled” in the spatial domain, are 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.