Constraint Energy Minimizing Generalized Multiscale Finite Element Method for High-Contrast Linear Elasticity Problem

In this paper, we consider the offline and online Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for high-contrast linear elasticity problem. Offline basis construction starts with an auxiliary multiscale space by solving local spectral problems. We select eigenfunctions that correspond to a few small eigenvalues to form the auxiliary space. Using the auxiliary space, we solve a constraint energy minimization problem to construct offline multiscale spaces. The minimization problem is defined in the oversampling domain, which is larger than the target coarse block. To get a good approximation space, the oversampling domain should be large enough. We also propose a relaxed minimization problem to construct multiscale basis functions, which will yield more accurate and robust solution. To take into account the influence of input parameters, such as source terms, we propose the construction of online multiscale basis and an adaptive enrichment algorithm. We provide extensive numerical experiments on 2D and 3D models to show the performance of the proposed method.     

Non-Matching Grids for a Flexible Discretization in Computational Acoustics

Flexible discretization techniques for the approximative solution of coupled wave propagation problems are investigated. In particular, the advantages of using non-matching grids are presented, when one subregion has to be resolved by a substantially finer grid than the other subregion. We present the non-matching grid technique for the case of a mechanical-acoustic coupled as well as for acoustic-acoustic coupled systems. For the first case, the problem formulation remains essentially the same as for the matching situation, while for the acoustic-acoustic coupling, the formulation is enhanced with Lagrange multipliers within the framework of Mortar Finite Element Methods. The applications will clearly demonstrate the superiority of the Mortar Finite Element Method over the standard Finite Element Method both concerning the flexibility for the mesh generation as well as the computational time.     

An Adjoint State Method for Numerical Approximation of Continuous Traffic Congestion Equilibria

The equilibrium metric for minimizing a continuous congested traffic model is the solution of a variational problem involving geodesic distances. The continuous equilibrium metric and its associated variational problem are closely related to the classical discrete Wardrop's equilibrium. We propose an adjoint state method to numerically approximate continuous traffic congestion equilibria through the continuous formulation. The method formally derives an adjoint state equation to compute the gradient descent direction so as to minimize a nonlinear functional involving the equilibrium metric and the resulting geodesic distances. The geodesic distance needed for the state equation is computed by solving a factored eikonal equation, and the adjoint state equation is solved by a fast sweeping method. Numerical examples demonstrate that the proposed adjoint state method produces desired equilibrium metrics and outperforms the subgradient marching method for computing such equilibrium metrics.     

Generalized Multiscale Finite Element Methods. Nonlinear Elliptic Equations

In this paper we use the Generalized Multiscale Finite Element Method (GMsFEM) framework, introduced in [26], in order to solve nonlinear elliptic equations with high-contrast coefficients. The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem formulation. With this convention, we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin (CG) or discontinuous Galerkin (DG) global formulations. Here, we use Symmetric Interior Penalty Discontinuous Galerkin approach. Both methods yield a predictable error decline that depends on the respective coarse space dimension, and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples.     

3D simulation of needle-tissue interaction with application to prostate brachytherapy.

This paper presents a needle-tissue interaction model that is a 3D extension of prior work based on needle and tissue models discretized using the Finite Element Method. The use of flexible needles necessitates remeshing the tissue during insertion, since simple mesh-node snapping to the tip can be detrimental to the simulation. In this paper, node repositioning and node addition are the two methods of mesh modification examined for coarse meshes. Our focus is on numerical approaches for fast implementation of these techniques. Although the two approaches compared, namely the Woodbury formula (matrix inversion lemma) and the boundary condition switches, have the same computational complexity, the Woodbury formula is shown to perform faster due to its cache-efficient order of operations. Furthermore, node addition is applied in constant time for both approaches, whereas node repositioning requires longer and variable computational times. A method for rendering the needle forces during simulated insertions into a 3D prostate model has been implemented. Combined with a detailed anatomical segmentation, this will be useful in teaching the practice of prostate brachytherapy. Issues related to discretization of such coupled (e.g., needle-tissue) models are also discussed.     

Robust open‐loop Nash equilibria in the noncooperative LQ game revisited

This paper reconsiders existence of worst‐case Nash equilibria in noncooperative multi‐player differential games, this, within an open‐loop information structure. We show that these equilibria can be obtained by determining the open‐loop Nash equilibria of an associated differential game with an additional initial state constraint. For the special case of linear‐quadratic differential games, we derive both necessary and sufficient conditions for solvability of the finite planning horizon problem. In particular, we demonstrate that, unlike in the standard linear‐quadratic differential game setting, uniqueness of equilibria may fail to hold. A both necessary and sufficient condition under which there is a unique equilibrium is provided. A sufficient existence condition for a unique equilibrium is derived in terms of a Riccati differential equation. Consequences for control policies are demonstrated in a simple debt stabilization game. © 2016 The Authors. Optimal Control Applications and Methods published by John Wiley & Sons, Ltd.     

Multiscale Finite Element Methods for Flows on Rough Surfaces

In this paper, we present the Multiscale Finite Element Method (MsFEM) for problems on rough heterogeneous surfaces. We consider the diffusion equation on oscillatory surfaces. Our objective is to represent small-scale features of the solution via multiscale basis functions described on a coarse grid. This problem arises in many applications where processes occur on surfaces or thin layers. We present a unified multiscale finite element framework that entails the use of transformations that map the reference surface to the deformed surface. The main ingredients of MsFEM are (1) the construction of multiscale basis functions and (2) a global coupling of these basis functions. For the construction of multiscale basis functions, our approach uses the transformation of the reference surface to a deformed surface. On the deformed surface, multiscale basis functions are defined where reduced (1D) problems are solved along the edges of coarse-grid blocks to calculate nodal multiscale basis functions. Furthermore, these basis functions are transformed back to the reference configuration. We discuss the use of appropriate transformation operators that improve the accuracy of the method. The method has an optimal convergence if the transformed surface is smooth and the image of the coarse partition in the reference configuration forms a quasiuniform partition. In this paper, we consider such transformations based on harmonic coordinates (following H. Owhadi and L. Zhang [Comm. Pure and Applied Math., LX(2007), pp. 675–723]) and discuss gridding issues in the reference configuration. Numerical results are presented where we compare the MsFEM when two types of deformations are used for multiscale basis construction. The first deformation employs local information and the second deformation employs a global information. Our numerical results show that one can improve the accuracy of the simulations when a global information is used.     

A generalized Nash equilibrium approach for optimal control problems of autonomous cars

We consider optimal control problems with ordinary differential equations that are coupled by shared, possibly nonconvex, constraints. For these problems, we use the generalized Nash equilibrium approach and provide a reformulation of normalized Nash equilibria as solutions to a single optimal control problem. By this reformulation, we are able to prove existence, and in some settings, exploiting convexity properties, we also get a limited number or even uniqueness of the normalized Nash equilibria. Then, we use our approach to discuss traffic scenarios with several autonomous vehicles, whose dynamics is described through differential equations, and the avoidance of collisions couples the optimal control problems of the vehicles. For the solution to the discretized problems, we prove strong convergence of the states and weak convergence of the controls. Finally, using existing optimal control software, we show that the generalized Nash equilibrium approach leads to reasonable results for a crossing scenario with different vehicle models.     

A Three-Level Multi-Continua Upscaling Method for Flow Problems in Fractured Porous Media

Traditional two level upscaling techniques suffer from a high offline cost when the coarse grid size is much larger than the fine grid size one. Thus, multilevel methods are desirable for problems with complex heterogeneities and high contrast. In this paper, we propose a novel three-level upscaling method for flow problems in fractured porous media. Our method starts with a fine grid discretization for the system involving fractured porous media. In the next step, based on the fine grid model, we construct a nonlocal multi-continua upscaling (NLMC) method using an intermediate grid. The system resulting from NLMC gives solutions that have physical meaning. In order to enhance locality, the grid size of the intermediate grid needs to be relatively small, and this motivates using such an intermediate grid. However, the resulting NLMC upscaled system has a relatively large dimension. This motivates a further step of dimension reduction. In particular, we will apply the idea of the Generalized Multiscale Finite Element Method (GMsFEM) to the NLMC system to obtain a final reduced model. We present simulation results for a two-dimensional model problem with a large number of fractures using the proposed three-level method.     

Preliminary Design and Optimization of an ECC Blood Pump by Means of a Parametric Approach

Abstract: This study concerns the development of an analytical parametric model of a centrifugal disk pump. The advantage of this kind of approach is to have an adaptable tool as a first step for the design of a pump device. The method allows the evaluation of the velocity profiles and the shear stresses within the impeller disks in the flow domain along with the performance of the device in terms of torque, mechanical power, power loss, head-flow performance, pump efficiency, and hemolytic index. Some simplifying hypotheses are assumed: steady state condition, laminar flow, Newtonian and incompressible fluid. The radial velocity profiles are assumed to be uniform and the flow cross-sectional area is assumed to be constant along the radius. The influence of the housing and secondary flows caused by recirculation are neglected. To test the approach reliability, the model was used to simulate a pump with the following characteristics: an external and internal radius of 50 mm and 5 mm, respectively, and a channel height of 2.5-0.25 mm ( h ) from inlet to outlet section. The angular velocity ω was varied in the range 500-3,000 rpm. The flow rate has been varied from 1 to 5 L/min. The results show that when the flow rate is increased, head performances obtained using this pump model vary from 411 to 100 mm Hg, and its efficiency varies from 48 to 15%. A parallel simulation has been carried out by means of a Finite Element Method model with an angular velocity equal to 2,000 rpm. The resulting comparison shows a good agreement between the results of these approaches. This occurrence indicates that the analytical approach is a valid and simple tool in defining pump design.     

3D reconstruction of the proximal femur with low-dose digital stereoradiography.

OBJECTIVE: Accurate three-dimensional (3D) geometry of the proximal femur may be helpful for fracture risk evaluation, as well as for planning and assisting surgical procedures. The purpose of this study was to apply and validate a stereoradiographic 3D reconstruction method on the proximal femur from radiographic contours identified on bi-planar radiographs. MATERIALS AND METHODS: Twenty-five excised non-pathologic proximal femurs were investigated using a low-dose digital radiographic device. Three-dimensional personalized models were reconstructed using the Non-Stereo Corresponding Contours (NSCC) algorithm. Three-dimensional CT-scan reconstructions were defined as geometric references for the comparison protocol, in order to assess the accuracy and reproducibility of the personalized 3D stereoradiographic reconstructions. In addition, the reliability of a set of 3D parameters obtained from stereoradiographic models was evaluated. RESULTS: This study demonstrated the validity of the NSCC method when applied to the proximal femur, with good results for accuracy (mean error = 0.7 mm) and reproducibility (Wilcoxon test: p > 0.28). Moreover, a precision study for the set of 3D parameters yielded a coefficient of variation lower than 5%. CONCLUSIONS: Once this approach has been validated in vivo, it should find multiple applications in therapeutic fields (e.g., for surgical planning, computer assisted surgery, etc.), as well as in diagnostic contexts (e.g., equilibrium studies or osteoporosis fracture risk assessment).     

Ship differential game approach for multiple players: Stackelberg security games

Casting the problem as a pursuit-evasion behavior in continuous time, this paper addresses a class of multiplayer ship differential Stackelberg security game. We are concerned with the conditions under which the defenders can capture the attackers. We represent the Stackelberg game as a Nash game for relaxing the interpretation of the noncooperative solution and the equilibrium selection problem. The weights of the players for the Nash solution are determined by their role in the Stackelberg game. The defenders try to minimize the capture condition. The attackers, knowing that they are being pursued by defenders, try to maximize the capture condition and minimize the distance to a certain target. For computing the equilibrium of the game, we employ a saddle-point method approach. The method consists of two half-steps iterated procedure where the functional of the game decrease and finally converges to an equilibrium point. We present the analysis of the convergence. Finally, we give a numerical example to illustrate the effectiveness and usefulness of our approach.     

Interior Penalty Discontinuous Galerkin Based Isogeometric Analysis for Allen-Cahn Equations on Surfaces

We propose a method that combines Isogeometric Analysis (IGA) with the interior penalty discontinuous Galerkin (IPDG) method for solving the Allen-Cahn equation, arising from phase transition in materials science, on three-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted at patch level, i.e., we employ the standard IGA within each patch, and employ the IPDG method across the patch interfaces. IGA is very suitable for solving Partial Differential Equations (PDEs) on (3D) surfaces and the IPDG method is used to glue the multiple patches together to get the right solution. Our method takes advantage of both IGA and the IPDG method, which allows us to design a superior semi-discrete (in time) IPDG scheme. First and most importantly, the time-consuming mesh generation process in traditional Finite Element Analysis (FEA) is no longer necessary and refinements, including h-refinement and p-refinement which both maintain the original geometry, can be easily performed at any level. Moreover, the flexibility of the IPDG method makes our method very easy to handle cases with non-conforming patches and different degrees across the patch interfaces. Additionally, the geometrical error is eliminated (for all conic sections) or significantly reduced at the beginning due to the geometric flexibility of IGA basis functions, especially the use of multiple patches. Finally, this method can be easily formulated and implemented. We present our semi-discrete IPDG scheme after generally describe the problem, and then briefly introduce the time marching method employed in this paper. Theoretical analysis is carried out to show that our method satisfies a discrete energy law, and achieves the optimal convergence rate with respect to the $L^2$ norm. Furthermore, we propose an elliptic projection operator on (3D) surfaces and prove an approximation error estimate which are vital for us to obtain the error estimate in the $L^2$ norm. Numerical tests are given to validate the theory and gauge the good performance of our method.     

A Finite Volume Method for the Multi Subband Boltzmann Equation with Realistic 2D Scattering in Double Gate MOSFETs

We propose a deterministic solver for the time-dependent multi-subband Boltzmann transport equation (MSBTE) for the two dimensional (2D) electron gas in double gate metal oxide semiconductor field effect transistors (MOSFETs) with flared out source/drain contacts. A realistic model with six-valleys of the conduction band of silicon and both intra-valley and inter-valley phonon-electron scattering is solved. We propose a second order finite volume method based on the positive and flux conservative (PFC) method to discretize the Boltzmann transport equations (BTEs). The transport part of the BTEs is split into two problems. One is a 1D transport problem in the position space, and the other is a 2D transport problem in the wavevector space. In order to reduce the splitting error, the 2D transport problem in the wavevector space is solved directly by using the PFC method instead of splitting into two 1D problems. The solver is applied to a nanoscale double gate MOSFET and the current-voltage characteristic is investigated. Comparison of the numerical results with ballistic solutions show that the scattering influence is not ignorable even when the size of a nanoscale semiconductor device goes to the scale of the electron mean free path.     

A Bilinear Immersed Finite Volume Element Method for the Diffusion Equation with Discontinuous Coefficient

This paper is to present a finite volume element (FVE) method based on the bilinear immersed finite element (IFE) for solving the boundary value problems of the diffusion equation with a discontinuous coefficient (interface problem). This method possesses the usual FVE method's local conservation property and can use a structured mesh or even the Cartesian mesh to solve a boundary value problem whose coefficient has discontinuity along piecewise smooth nontrivial curves. Numerical examples are provided to demonstrate features of this method. In particular, this method can produce a numerical solution to an interface problem with the usual O(h²) (in L² norm) and O(h) (in H¹ norm) convergence rates.     

颅骨缺损的个性化修复

目的 探索修复颅骨缺损的一种新型材料和方法。方法 利用三维重建技术和快速自动成型技术，采用医用钛预制成个性化钛植入体修复颅骨缺损。结果 自2001年以来，采用此法完成9例颅骨缺损的修复，随访3～12个月，效果良好。结论 颅骨缺损的个性化修复具有操作简单、并发症少以及修复精确、良好的生物相容性和生物力学等优点，具有推广应用的前景。     

A Well-Balanced Gas Kinetic Scheme for Navier-Stokes Equations with Gravitational Potential

The hydrostatic equilibrium state is the consequence of the exact balance between hydrostatic pressure and external force. Standard finite volume cannot keep this balance exactly due to their unbalanced truncation errors. In this study, we introduce an auxiliary variable which becomes constant at isothermal hydrostatic equilibria and propose a well-balanced gas kinetic scheme for the Navier-Stokes equations. Through reformulating the convection term and the force term via the auxiliary variable, zero numerical flux and zero numerical source term are enforced at the hydrostatic equilibrium state instead of the balance between hydrostatic pressure and external force. Several problems are tested to demonstrate the accuracy and the stability of the new scheme. The results confirm that, the new scheme can preserve the exact hydrostatic solution. The small perturbation riding on hydrostatic equilibria can be calculated accurately. More importantly, the new scheme is capable of simulating the process of converging towards hydrostatic equilibria from a highly unbalanced initial condition. The ultimate state of zero velocity and constant temperature is achieved up to machine accuracy. As demonstrated by the numerical experiments, the current scheme is very suitable for small amplitude perturbation and long time running under gravitational potential.     

Total anorectal reconstruction with an artificial bowel sphincter: Report of five cases with a minimum follow‐up of 6 months

Background The artificial bowel sphincter (Acticon ABS – American Medical Systems, Minneapolis, MN, USA) has been proposed as a treatment for patients with faecal incontinence. The good results achieved with this procedure encouraged us to utilize this device for reconstruction of patients who previously underwent an abdominoperineal resection (APR). Method Between 1999 and 2000 we implanted the ABS in five patients undergoing an APR. One patient was male and four female, the mean age was 51.3 years. Three patients had been operated on for rectal cancer, one for rectal agenesia and one for a giant benign tumour of the pelvis. Results The length of follow up ranged from 6 to 22 months. Manometry assessed a basal pressure with the ABS cuff inflated between 58 and 62.2 mmHg. All but one achieved a good grade of continence with a Wexner score range between 3 and 9. A certain degree of impaired evacuation occurred in two patients but, with adequate training, this improved and did not affect patient satisfaction. Conclusion The ABS is a good option for reconstruction of patients previously treated with an APR. As compared to electrostimulated graciloplasty the ABS technique seems to be easier to perform and more acceptable for the patients, although the cost of the device is still high.     

New dosimetric approach for multidimensional dose evaluation in gamma knife radiosurgery. Technical note

During the past two decades, the progress in computerized treatment planning systems has led to more accurate imaging and therapy by using the gamma knife, especially with the smallest collimators (4 mm). However, the ionization chambers that have been used to calibrate the gamma knife are not useful with the smallest collimators because the chambers are too big compared with the irradiated volume. Therefore, it is important to develop more suitable dosimeters. This study proposes a new dosimeter method. The FriXyGel method proposed here is based on a phantom dosimeter, an acquisition chain, and dedicated software. This dosimeter uses an agarose gel into which a ferrous sulphate solution (Fricke solution) and a metal ion indicator (xylenol orange) are incorporated. The absorbed dose is detected through measurements of visible light transmission, imaged by means of a charge-coupled device camera provided with a suitable optical filter. Gel layers are imaged before and after irradiation, and the differences in light absorption are related to the absorbed dose. By choosing convenient thickness of gel layers and by building up a phantom with different gel slices, it is possible to obtain a three-dimensional (3D) representation of the absorbed dose. The final 3D representation is reached after several mathematical processes have been applied to the images. The first step identifies and reduces all factors that could alter the original data, such as nonuniformity in illumination. Then, after calibration procedures, it is possible to obtain absorbed dose values and to discover their 3D representation. This goal has been reached by developing appropriate software that performs all the calculations necessary for spatial representation routines and prompt comparison with theoretical calculations.     

Use of different buffers in peritoneal dialysis

A buffer is included in the peritoneal dialysis solution in order to offset the hydrogen ions normally produced during the metabolic processes. Nowadays, the buffer used is lactate, and its concentration in conventional peritoneal dialysis fluids is 35 or 40 mmol/L. Despite the general thought that peritoneal dialysis adequately corrects uremic acidosis, several studies have demonstrated that more than 50% of patients present mild to moderate acidosis with the solution containing 35 mmol/L of lactate, although with a 40 mmol/L solution this percentage decreases, a substantial number of patients still remain acidotic. This acid-base derangement is characterized by a normal pH and a below-normal plasma bicarbonate concentration, although the external body base balance is in equilibrium. There is evidence that this condition contributes to uremic osteodystrophy and has a detrimental effect on protein metabolism. Conventional solutions also affect mesothelial cell viability and local leukocyte function and have potential systemic effects such as the impairment of cellular redox state. New solutions containing pure bicarbonate or a mixture of bicarbonate and lactate have recently been investigated. A bicarbonate solution containing 34 mmol/L significantly increased plasma bicarbonate levels as compared with the lactate 35 mmol/L solution. It has been demonstrated that bicarbonate solutions have better biocompatibility than the lactate buffered solution and substantially reduce abdominal discomfort experienced by a certain percentage of patients during the solution infusion. These studies demonstrated that the bicarbonate-buffered CAPD solution is safe, well-tolerated, and does not present any, even potential, side effects. Thus, it seems reasonable to consider the bicarbonate buffered solution the standard instead of the alternative, and it might entirely replace lactate as buffer in peritoneal dialysis fluid.