Applications of Automatic Mesh Generation and Adaptive Methods in Computational Medicine

Projection-based embedded discrete fracture models (pEDFMs) are proven effective for modeling flow barrier effects of high-conductivity or impermeable fractures. However, local grid refinements are still needed to improve the accuracy of simulation in flow areas near fractures. In recent years, adaptive grid refinement techniques have received a lot of attention for dealing with highly heterogeneous and fractured models. But few of them are capable of EDFMs. In this paper, an adaptive grid refinement method under flow-based EDFMs (fEDFMs) is proposed for fractured models. The method starts from an fEDFM model which is built by a new technique of transmissibility modification by introducing an artificial pseudo-steady flow near fractures. Adaptive grid refinement and coarsening procedures are designed under an adaptive criterion based on both the fracture distribution and flow solutions. A flow-based upscaling procedure is adopted to form transmissibilities of the hybrid grids and the solution is mapped from the former grid system. The adaptive grid refinement method is applied in a validation case and a real field case, respectively. In each case, comparisons are made between the simulation results of the proposed adaptive grid refinement models and traditional uniform pEDFMs. Besides, comparisons are also made with the overall fine-scale models which serve as the reference models. The comparisons show that the numerical results of the proposed models have a better match to that of the reference models. And it is proven that the approach is more robust when applied to more general flow scenarios with extremely high or completely sealed fractures which could have a great impact on the flow. The proposed method aims to improve the accuracy of numerical simulation for fractured reservoirs.

Polygonal finite element provides a great flexibility in mesh generation of crack propagation problems where the topology of the domain changes significantly. However, the control of the discretization error in such problems is a main concern. In this paper, a polygonal-FEM is presented in modeling of crack propagation problems via an automatic adaptive mesh refinement procedure. The adaptive mesh refinement is accomplished based on the Zienkiewicz–Zhu error estimator in conjunction with a weighted SPR technique. Adaptive mesh refinement is employed in some steps for reduction of the discretization error and not for tracking the crack. In the steps that no adaptive mesh refinement is required, local modifications are applied on the mesh to prevent poor polygonal element shapes. Finally, several numerical examples are analyzed to demonstrate the efficiency, accuracy and robustness of the proposed computational algorithm in crack propagation problems.

Summary An adaptive three-dimensional mesh refinement method based on the law of mass conservation has been introduced and tested using some analytical velocity fields as accurate in identifying singular point, asymptotic plane and drawing closed streamlines. This paper further investigates the adaptive mesh refinement method using a velocity field that is due to a uniform strain and a point vortex. Similar results have been obtained.

An ANN-based element extraction method for automatic mesh generation

We consider variationally consistent discretization schemes for mechanical contact problems. Most of the results can also be applied to other variational inequalities, such as those for phase transition problems in porous media, for plasticity or for option pricing applications from finance. The starting point is to weakly incorporate the constraint into the setting and to reformulate the inequality in the displacement in terms of a saddle-point problem. Here, the Lagrange multiplier represents the surface forces, and the constraints are restricted to the boundary of the simulation domain. Having a uniform inf-sup bound, one can then establish optimal low-ordera prioriconvergence rates for the discretization error in the primal and dual variables. In addition to the abstract framework of linear saddle-point theory, complementarity terms have to be taken into account. The resulting inequality system is solved by rewriting it equivalently by means of the non-linear complementarity function as a system of equations. Although it is not differentiable in the classical sense, semi-smooth Newton methods, yielding super-linear convergence rates, can be applied and easily implemented in terms of a primal–dual active set strategy. Quite often the solution of contact problems has a low regularity, and the efficiency of the approach can be improved by using adaptive refinement techniques. Different standard types, such as residual- and equilibrated-baseda posteriorierror estimators, can be designed based on the interpretation of the dual variable as Neumann boundary condition. For the fully dynamic setting it is of interest to apply energy-preserving time-integration schemes. However, the differential algebraic character of the system can result in high oscillations if standard methods are applied. A possible remedy is to modify the fully discretized system by a local redistribution of the mass. Numerical results in two and three dimensions illustrate the wide range of possible applications and show the performance of the space discretization scheme, non-linear solver, adaptive refinement process and time integration.

New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

Higher order optimization and adaptive numerical solution for optimal control of monodomain equations in cardiac electrophysiology

Spurious interface and boundary behaviour beyond physical solutions in lattice Boltzmann schemes

Effects of tip slots on the aerodynamic characteristics of helicopter rotor were investigated numerically by solving three-dimensional Navier-Stokes equations based on unstructured overset grids algorithm. Improved delayed detached eddy simulation (IDDES) based on the Spalart-Allmaras turbulence model and adaptive grid refinement technique were employed. Several slots in the rotor blade tip were designed on the base of Caradonna-Tung rotor to study the effect of tip slots. Numerical results show that tip slots are able to introduce the airflow from the leading edge and turn it in the spanwise direction to be ejected out of the face at the rotor blade tip, which can reduce the strength of the rotor blade tip vortex and accelerate the dissipation process. Although tip slots may lead to the decrease of airfoils' lift coefficient at the root of the rotor blade, it can increase the lift coefficient of airfoils at the rotor blade tip, so the lift of the rotor with tip slots is almost the same as that of the rotor without tip slots. In addition, tip slots can also reduce the intensity of the tip shock wave, which is beneficial to reduce the wave drag of the rotor.

Atmospheric, laminar, stoichiometric, premixed hydrogen-air flames in a diverging channel are investigated by means of Computational Fluid Dynamics. This configuration has been recently used in a series of experimental investigations to determine the burning velocities and quenching distances for premixed flames of different fuels. The purpose of the present investigation is the validation of the prediction procedures for the burning speeds and quenching distances for hydrogen flames by comparing them with these measurements. Global and detailed reaction mechanisms are applied to describe the combustion process. For assuring an adequately fine resolution of the flame fronts, adaptive grid refinement techniques are applied. A reasonable agreement is observed with the experiments, where the detailed and global mechanisms are slightly overpredicting and underpredicting the quenching distance, respectively.

Head-on collision of droplets consisting of different liquids for a wide range of Weber numbers between 25 and 430 is studied numerically using the volume of fluid (VOF) methodology and by adopting an axisymmetric approximation. An adaptive local grid refinement technique is used to capture with precision the liquid–air interface at significantly reduced computational cost relative to a uniformly dense grid. The conditions setting the boundary between permanent coalescence and reflexive separation are defined; in addition, useful relationships determining the ligament’s elongation, the corresponding time of its breakup, the number of satellite droplets formed, the maximum droplet deformation, and the viscous dissipation coefficient are presented. Furthermore, the present study reveals the dependence of collision characteristics not only on Weber number, but also either on Reynolds, capillary, or Ohnesorge numbers.

Adaptive in-time local grid refinement techniques use multilevel local discretizations designed to achieve local accuracy. The changing nature of the matrix structure of the linear systems arising from the multilevel local discretizations requires flexible approximate factorizations that focus on local components and coordinate their interaction. The solution of these composite grid systems with Krylov solvers is considered. The selection of an adequate preconditioner is crucial. The incomplete Cholesky (IC) factorization of the composite matrix and the inexact BEPS preconditioner are two such potential preconditioners. The inexact BEPS preconditioner can be constructed and applied with significantly more flexibility than the IC factorizations of the composite multilevel grid matrix. An extension of the IC factorizations for matrices arising from discretizations of self-adjoint PDEs on multilevel composite grids is proposed. The resulting factorization is referred to as the incomplete multilevel Cholesky (IMC) factorization. The IMC factorization is spectrally equivalent to the IC factorization of the matrix of the composite grid system. IMC factorization can be constructed with the same flexibility as the inexact BEPS preconditioner. The application of IMC factorization is achieved via a multilevel LLT-cycle consisting of a forward elimination pass proceeding downward on the grids from fine to coarse followed by an reverse-order upward back substitution pass. The application of multilevel factorization as a preconditioner requires roughly one-half as many operations as the inexact BEPS preconditioner. The numerical results illustrate the potential of the method.

A numerical investigation of central binary collision of droplets

Much attention has been focused on the problem of mesh generation because of its role as a bottleneck in the finite element analysis. One solution is to develop an efficient adaptive scheme. In adaptive mesh generation, accurate error estmation is indispensable, but it is also important to reflect the calculated result of the estimation exactly in the next step of refinement. In this paper, a new automatic adaptive mesh generation scheme based upon the technique of arbitrarily distributed random numbers is proposed. The steps outlining this scheme are as follows : (1) calculate the distribution of the desired node density from error estimation for a previous finite element solution (2) generate nodes according to the calculated node density distribution (3) construct elements by connecting the nodes. The generation of random numbers according to a given distribution is widely used in Monte Carlo simulation and this technique is applied to generate nodes of the desired density. The applicability of this scheme is examined in some examples. The decrease of the relative error obtained by iterative adaptive remeshing shows the superiority of the proposed method

The ability to relate data collected on the body surface to the electrical activity on the epicardial surface is of great importance in noninvasive detection and localization of many diseases of the heart. The computation of the potential distribution on the body surface from the potential distribution on the surface of the hearty (epicardial potentials) constitutes the forward problem in electrocardiography. Conversely, the problem of the reconstruction of epicardial potentials from body surface potentials is called the inverse problem in electrocardiography. Modeling and analysis related to these two problems are presented. >