Anisotropic mesh adaptation with Riemannian metrics has proven effective for generating straight-sided meshes with anisotropy induced by the geometry of interest and/or the resolved physics. Within the continuous mesh framework, anisotropic meshes are thought of as discrete counterparts to Riemannian metrics. Ideal, or unit, simplicial meshes consist only of simplices whose edges exhibit unit or quasi-unit length with respect to a given Riemannian metric. Recently, mesh adaptation with high-order (i.e., curved) elements has grown in popularity in the meshing community, as the additional flexibility of high-order elements can further reduce the approximation error. However, a complete and competitive methodology for anisotropic and high-order mesh adaptation is not yet available. The goal of this paper is to address a key aspect of metric-based high-order mesh adaptation, namely, the adequacy between a Riemannian metric and high-order simplices. This is done by extending the notions of unit simplices and unit meshes, central to the continuous mesh framework, to high-order elements. The existing definitions of a unit simplex are reviewed, then a broader definition involving Riemannian isometries is introduced to handle curved and high-order simplices. Similarly, the notion of quasi-unitness is extended to curved simplices to tackle the practical generation of high-order meshes. Proofs of concept for unit and (quasi-)isometric meshes are presented in two dimensions.

This paper describes a framework for the generation of $$P^2$$ isoparametric meshes in two dimensions. It is an extension of existing metric-based methods for high-order straight-sided mesh adaptation to curved meshes. Starting with an interpolation error estimate for curved trajectories, we compute an optimal metric field using a generalization of the log-simplex algorithm for high-order metrics. A straight quasi-unit mesh is generated in a frontal approach, then the edges are curved to minimize their length in the chosen metric. Convergence studies are performed on simple test cases. In particular, optimal convergence rates (third order) are reached even when the edges curvature decreases at the same rate as the edge length.

This paper presents a sequence of variable time step deferred correction (DC) methods constructed recursively from the second-order backward differentiation formula (BDF2) applied to the numerical solution of initial value problems for first-order ordinary differential equations (ODE). The sequence of corrections starts with the BDF2 then considered as DC2. We prove that this improvement from a p -order solution (DC p ) results in a p+1 -order accurate solution (DC p+1 ). This one-order increment in accuracy holds for the least stringent BDF2 0-stability conditions. If we introduce additional requirements for the ratio of consecutive variable time step sizes, then the order increment is 2, allowing a direct transition from DC p to DC p+2 . These requirements include the constant time step DC p methods. We also prove that all these DC p methods are A-stable. We briefly discuss two other DC variants to illustrate how a proper transition from DC p to DC p+1 is critical to maintaining A-stability at all orders. Numerical experiments based on two manufactured (closed-form) solutions confirmed the accuracy orders of the DC p – for DC p , p=2,3,4,5 – both with constant or alternating time step sizes. We showed that the theoretical conditions required to obtain an increment of orders 1 and 2 are satisfied in practice. Finally, a test case shows that we can estimate the error on the DC p solution with the DC p+1 solution, and a last test case that our new methods maintain their order of accuracy for a stiff system.

The central aim of this paper is to use OpenFOAM for the assessment of mesh resolution requirements for large-eddy simulation (LES) of flows similar to the ones which occur inside the draft-tube of hydraulic turbines at off-design operating conditions. The importance of this study is related to the fact that hydraulic turbines often need to be operated over an extended range of operating conditions, which makes the investigation of fluctuating stresses crucial. Scale-resolving simulation (SRS) approaches, such as LES and detached-eddy simulation (DES), have received more interests in the recent decade for understanding and mitigating unsteady operational behavior of hydro turbines. This interest is due to their ability to resolve a larger part of turbulent flows. However, verification studies in LES are very challenging, since errors in numerical discretization, but also subgrid-scale (SGS) models, are both influenced by grid resolution. A comprehensive examination of the literature shows that SRS for different operating conditions of hydraulic turbines is still quite limited and that there is no consensus on mesh resolution requirement for SRS studies. Therefore, the goal of this research is to develop a reliable framework for the validation and verification of SRS, especially LES, so that it can be applied for the investigation of flow phenomena inside hydraulic turbine draft-tube and runner at their off-design operating conditions. Two academic test cases are considered in this research, a turbulent channel flow and a case of sudden expansion. The sudden expansion test case resembles the flow inside the draft-tube of hydraulic turbines at part load. In this study, we concentrate on these academic test cases, but it is expected that hydraulic turbine flow simulations will eventually benefit from the results of the current research. The results show that two-point autocorrelation is more sensitive to mesh resolution than energy spectra. In addition, for the case of sudden expansion, the mesh resolution has a tremendous effect on the results, and, so far, we have not capture an asymptotic converging behavior in the results of Root Mean Square (RMS) of velocity fluctuations and two-point autocorrelation. This case, which represents complex flow behavior, needs further mesh resolution studies.

This paper presents a cost-effective adaptive remeshing algorithm for constructing model- consistent wall functions for low-Reynods number models of turbulence. Traditional wall functions are obtained by developing algebraic expressions for the dependent variables (u, k, ε) in 1D plane turbulent Couette flow. However, in general, closed form solutions to the Couette flow coupled system of equations can only be obtained by invoking additional approximations whose impact on solution accuracy is difficult to quantify. Numerical solutions of the 1D Couette flow avoids this problem. The resulting tabulated wall functions are fully compatible and consistent with the turbulence model. We have opted for a finite element method based on adaptive remeshing because it yields highly accurate boundary conditions with a small number of optimally placed nodes.

In this paper, the efficiency in mesh updating (r-adaptivity) of the Transfinite Mean value Interpolation (TMI) and its generalization (k-TMI) are compared on three standardized problems to the well-known Inverse Distance Weighted interpolation (IDW) and Radial Basis Function interpolation (RBF) for unstructured data points and the new k-Transfinite Barycentric Interpolation (k-TBI) for structured data points such as, for instance, curves or surfaces in 3D. This is achieved by introducing a dynamical version of these interpolations via an ordinary differential equation that can be solved by standard ODE methods that are more economical than, for instance, solving vector partial differential equations as in the pseudo-solid method.A review of the very recent mathematical foundations of the k-TMI and k-TBI constructed from the function alone (standard) or from the function and its derivatives (enhanced) is provided in the first part of the paper.

This work highlights a new mesh adaption procedure which takes action locally. The procedure is specially designed for the simulation of unsteady flows. The methodology is explained in a two-dimensional context but could be extended to tackle three-dimensional problems. This approach is intended to be an interesting alternative to techniques based on local mesh subdivision or fusion. The method uses the gradient recovery technique of Zhu and Zienkiewicz to estimate the spatial error, and an advancing front meshing tool to mesh the computational domain. The elements removed from the mesh, denoted seeds, are identified by their size variation predicted by the mesh adaption method. The mesh updates are triggered by several stopping criteria which also suspend the time-integration. The process is therefore completely automatic. The work presented here was carried out within the framework of the finite element method.

This paper presents a new monolithic formulation targeted at the simulation of fluid-structure interaction. It combines the dynamic version of the recent k-TMI extension of the Transfinite Mean Value Interpolation (TMI) of Dyken and Floater with the ALE formulation of the Navier-Stokes equations for moving finite element meshes along with the Lagrange multiplier method to accurately predict parietal shear. This new integrated k-TMI/ALE formulation is stable for the extreme case of zero mass ratios.

Purpose This paper aims to focus on characterization of interactions between hp-adaptive time-integrators based on backward differentiation formulas (BDF) and adaptive meshing based on Zhu and Zienkiewicz error estimation approach. If mesh adaptation only occurs at user-supplied times and results in a completely new mesh, it is necessary to stop the time-integration at these same times. In these conditions, one challenge is to find an efficient and reliable way to restart the time-integration. The authors investigate what impact grid-to-grid interpolation errors have on the relaunch of the computation. Design/methodology/approach Two restart strategies of the time-integrator were used: one based on resetting the time-step size h and time-integrator order p to default values (used in the initial startup phase), and another designed to restart with the time-step size h and order p used by the solver prior to remeshing. The authors also investigate the benefits of quadratically interpolate the solution on the new mesh. Both restart strategies were used to solve laminar incompressible Navier–Stokes and the Unsteady Reynolds Averaged Naviers-Stokes (URANS) equations. Findings The adaptive features of our time-integrators are excellent tools to quantify errors arising from the data transfer between two grids. The second restart strategy proved to be advantageous only if a quadratic grid-to-grid interpolation is used. Results for turbulent flows also proved that some precautions must be taken to ensure grid convergence at any time of the simulation. Mesh adaptation, if poorly performed, can indeed lead to losing grid convergence in critical regions of the flow. Originality/value This study exhibits the benefits and difficulty of assessing both spatial error estimates and local error estimates to enhance the efficiency of unsteady computations.

Stents are used in interventional cardiology in order to keep a diseased vessel open. New stents are coated with a medicinal agent that prevents the early reclosing caused by the proliferation of smooth muscle cells (SMC). In order to obtain the desired release kinetics for the SMC-controlling drug during the required therapeutic period, the current strategy focuses on biphasic or possibly polyphasic release from blends of degradable polymers. Blanchet and co-authors introduced a two-parameter ordinary differential equation to accurately characterize the release kinetics from the experimental release curves of Lao and co-authors for neat polymers and polymer blends under infinite sink conditions. From this Garon and Delfour constructed a three-dimensional semiempirical model for the flat polymeric film in a closed vial without fluid circulation where all the parameters can be measured in the laboratory. The object of this paper is the extension of that work to the three-dimensional modeling of drug release from the polymeric coating of a stent of arbitrary pattern to the wall/lumen of a curved blood vessel. The model also takes into account diffusion and metabolization in its wall and diffusion and blood circulation in its lumen. It is a practical tool to theoretically and numerically simulate the three-dimensional drug release from a thin polymeric coating to the aggregated wall and lumen of the blood vessel for design and evaluation.

In r-adaptivity, several methods are available to extend boundary motion analytically into the computational domain. We propose a new mesh deformation technique based on transfinite mean value interpolation (TMI). In its original version TMI is explicit, matrix-free, and linearly exact, i.e., a solid rotation of the whole mesh is accurate. The mesh updating efficiency and robustness of the TMI, inverse distance weighting method (IDW) and radial basis functions method (RBF) are compared.

Purpose This paper aims to focus on characterization of interactions between hp-adaptive time-integrators based on backward differentiation formulas (BDF) and adaptive meshing based on Zhu and Zienkiewicz error estimation approach. If mesh adaptation only occurs at user-supplied times and results in a completely new mesh, it is necessary to stop the time-integration at these same times. In these conditions, one challenge is to find an efficient and reliable way to restart the time-integration. The authors investigate what impact grid-to-grid interpolation errors have on the relaunch of the computation. Design/methodology/approach Two restart strategies of the time-integrator were used: one based on resetting the time-step size h and time-integrator order p to default values (used in the initial startup phase), and another designed to restart with the time-step size h and order p used by the solver prior to remeshing. The authors also investigate the benefits of quadratically interpolate the solution on the new mesh. Both restart strategies were used to solve laminar incompressible Navier-Stokes and the Unsteady Reynolds Averaged Naviers-Stokes (URANS) equations. Findings The adaptive features of our time-integrators are excellent tools to quantify errors arising from the data transfer between two grids. The second restart strategy proved to be advantageous only if a quadratic grid-to-grid interpolation is used. Results for turbulent flows also proved that some precautions must be taken to ensure grid convergence at any time of the simulation. Mesh adaptation, if poorly performed, can indeed lead to losing grid convergence in critical regions of the flow. Originality/value This study exhibits the benefits and difficulty of assessing both spatial error estimates and local error estimates to enhance the efficiency of unsteady computations.

Major moving biological fluids, or biofluids, are blood and air that cooperate to bring oxygen to the body’s cells and eliminate carbon dioxide produced by these cells. Blood is conveyed in a closed network composed of 2 circuits in series — the systemic and pulmonary circulation —, each constituted by arteries, capillaries, and veins, under the synchronized action of the left and right cardiac pumps, respectively. Air is successively inhaled from and exhaled to the atmosphere through the airway openings (nose and/or mouth). In the head, blood generates the cerebrospinal fluid in choroid plexi of all compartments of the ventricular system and receives it in arachnoid villi. Other biofluids are either secreted, such as bile from the liver and breast milk that both transport released substances with specific tasks, or excreted, such as urine from kidneys or sweat from skin glands that both convey useless materials and waste produced by the cell metabolism. In addition to the convective transport, peristalsis, which results from the radial contraction and relaxation of mural smooth muscles, propels the content of the lumen of the muscular bioconduit (e.g., digestive tract) in an anterograde direction. Blood circulation and air flow in the respiratory tract are widely explored because of their vital functions. Note that inhaled air is transported through the respiratory tract by two processes: convection down to bronchioles and diffusion down to pulmonary alveoli.1 Furthermore, investigations of these physiological flows in deformable bioconduits give rise to models such as the Starling resistance that themselves become object of new study fields in physics and mechanics (e.g., collapsible tubes) as well as of new developments in math- ematical modeling and scientific computing. Whereas fluid–structure interaction problems in aeronautics and civil engineering deal with materials of distinct properties, blood stream and vessel wall correspond to two domains of nearly equal physical properties, as both blood and vessels have densities close to that of water. New processing strategies must then be conceived.

Implantation of an artificial urinary sphincter (AUS) is the treatment of choice for managing severe stress urinary incontinence. This hydromechanical implantmimics a healthy sphincter by exerting a constant circumferential pressure around the urethra to close it and keep urine in the bladder. Common complications experienced with this device are urethral atrophy, erosion, and mechanical breakdown. Furthermore, AUS operation requires some dexterity resulting in difficulty and discomfort of use, and limiting the implantation in some patients. We present in this study a novel remote-controlled electromechanic AUS allowing rapid and effortless sphincter operation. Its design eliminates the classic manual pump, enhancing reliability, easing implantation in men and women, and achieving compatibility with already-implanted AUS. The device has been tested in vitro and ex vivo on fresh pig bladders. Different occlusive cuff pressure (OCP) ranges were employed and expected performance was obtained. Experimentation results and design challenges are reported and discussed here in.

Stress urinary incontinence is a complication that considerably affects patient quality of life. When conservative measures are not sufficient for recovering patient's continence, urologists suggest the implantation of an artificial urinary sphincter (AUS). The AMS 800 has been the gold standard AUS for more than 35 years now. This hydro mechanical device mimics the behavior of a healthy sphincter. However, it suffers from several limitations reported by medical studies and follow up. In this paper, we present our concept of the smart AUS. The latter corrects the limitations and extends the capabilities of current AUSs. We report here-in three different versions of the smart AUS. Each version integrates novel capabilities and unique features. Each device is presented and its experimentation results are detailed. Design challenges are discussed and technological barriers for the further development of actual AUSs are explained.