An Iterative Thresholding Method for the Minimum Compliance Problem

In this paper, we propose a simple energy decaying iterative thresholding algorithm to solve the two-phase minimum compliance problem. The material domain is implicitly represented by its characteristic function, and the problem is formulated into a minimization problem by the principle of minimum complementary energy. We prove that the energy is decreasing in each iteration. Two effective continuation schemes are proposed to avoid trapping into the local minimum. Numerical results on 2D isotropic linear material demonstrate the effectiveness of the proposed methods.     

A Hodge Decomposition Method for Dynamic Ginzburg–Landau Equations in Nonsmooth Domains — A Second Approach

In a general polygonal domain, possibly nonconvex and multi-connected (with holes), the time-dependent Ginzburg–Landau equation is reformulated into a new system of equations. The magnetic field $B$:=∇×A is introduced as an unknown solution in the new system, while the magnetic potential A is solved implicitly through its Hodge decomposition into divergence-free part, curl-free and harmonic parts, separately. Global well-posedness of the new system and its equivalence to the original problem are proved. A linearized and decoupled Galerkin finite element method is proposed for solving the new system. The convergence of numerical solutions is proved based on a compactness argument by utilizing the maximal $L^p$-regularity of the discretized equations. Compared with the Hodge decomposition method proposed in [27]_,the new method has the advantage of approximating the magnetic field B directly and converging for initial conditions that are incompatible with the external magnetic field. Several numerical examples are provided to illustrate the efficiency of the proposed numerical method in both simply connected and multi-connected nonsmooth domains. We observe that even in simply connected domains, the new method is superior to the method in [27] for approximating the magnetic field.     

On the Five-Moment Maximum Entropy System of One-Dimensional Boltzmann Equation

The maximum entropy moment system extends the Euler equation to nonequilibrium gas flows by considering higher order moments such as the heat flux. This paper presents a systematic study of the maximum entropy moment system of Boltzmann equation. We consider a hypothetical one-dimensional gas and study a five-moment model. A numerical algorithm for solving the optimization problem is developed to produce the maximum entropy distribution function from known moments, and the asymptotic behaviour of the system around the singular region known as the Junk’s line, as well as that near the boundary of the realizability domain is analyzed. Furthermore, we study the properties of the system numerically, including the behaviour of the system around the Maxwellian and within the interior of the realizability domain, and properties of its characteristic fields. Our studies show the higher order entropy-based moment models to differ significantly from the Euler equations. Much of this difference comes from the singularity near the Junk’s line, which would be removed if a truncation of the velocity domain is employed.     

Antiplane Wave Scattering from a Cylindrical Void in a Pre-Stressed Incompressible Neo-Hookean Material

An isolated cylindrical void is located inside an incompressible nonlinear-elastic medium whose constitutive behaviour is governed by a neo-Hookean strain energy function. In-plane hydrostatic pressure is applied in the far-field so that the void changes its radius and an inhomogeneous region of deformation arises in the vicinity of the void. We consider scattering from the void in the deformed configuration due to an incident field (of small amplitude) generated by a horizontally polarized shear (SH) line source, a distance r₀ (R₀) away from the centre of the void in the deformed (undeformed) configuration. We show that the scattering coefficients of this scattered field are unaffected by the pre-stress (initial deformation). In particular, they depend not on the deformed void radius a or distance r₀, but instead on the original void sizeA and original distance R₀.     

Gap metric–based model bank construction for wind turbine predictive control

This paper presents a novel model bank construction method for the multiple model predictive control of wind turbine system. The gap metric is used to measure the dynamic difference between the linearized models of the wind turbine system at different wind speed. Two algorithms are then proposed to divide the wind speed range in different operating regions. Meanwhile, a complete and nonredundant linear model bank is established to approximate the wind turbine system in the whole operating region. We take the robust model predictive control algorithm to design the local controller and utilize the wind speed as the switching criterion to combine the submodels. The simulation study on a 5‐MW wind turbine verifies the efficiency of the proposed method.     

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.     

Boundary Control Problems in Convective Heat Transfer with Lifting Function Approach and Multigrid Vanka-Type Solvers

This paper deals with boundary optimal control problems for the heat and Navier-Stokes equations and addresses the issue of defining controls in function spaces which are naturally associated with the volume variables by trace restriction. For this reason we reformulate the boundary optimal control problem into a distributed problem through a lifting function approach. The stronger regularity requirements which are imposed by standard boundary control approaches can then be avoided. Furthermore, we propose a new numerical strategy that allows solving the coupled optimality system in a robust way for a large number of unknowns. The optimality system resulting from a finite element discretization is solved by a local multigrid algorithm with domain decomposition Vanka-type smoothers. The purpose of these smoothers is to solve the optimality system implicitly over subdomains with a small number of degrees of freedom, in order to achieve robustness with respect to the regularization parameters in the cost functional. We present the results of some test cases where temperature is the observed quantity and the control quantity corresponds to the boundary values of the fluid temperature in a portion of the boundary. The control region for the observed quantity is a part of the domain where it is interesting to match a desired temperature value.     

Sagittal laxity in vivo after total knee arthroplasty

Introduction A stress arthrometry study of 77 knees undergoing total knee arthroplasty was performed to determine the difference in anteroposterior (AP) laxity between posterior cruciate ligament (PCL)-retaining (PCLR) and PCL-substituting (PCLS) prostheses using the Genesis I TKA.Materials and methods Fifty-three knees had PCLR and 24 had PCLS prostheses. The selected patients had successful arthroplasties after a minimum follow-up of 5 years. AP laxity was measured with a KT-2000 arthrometer (Medmetric, San Diego, CA, USA) using standard protocols.Results At 30° of flexion, there was no statistical difference in anterior (PCLR: 4.7 mm, PCLS: 4.5 mm), posterior (PCLR: 1.1 mm, PCLS: 0.7 mm), or total (PCLR: 5.8 mm, PCLS: 5.3 mm) displacement. At 75°, significant differences were seen in both anterior (PCLR: 3.3 mm, PCLS: 2.3 mm) and total (PCLR: 4.8 mm, PCLS: 3.4 mm) displacement (p=0.001 and p=0.009, respectively), although there was no statistical difference in posterior displacement (PCLR: 1.5 mm, PCLS: 1.1 mm).Conclusion The above values are considered the suitable degree of AP laxity in total knee arthroplasty for a satisfactory clinical outcome 5–9 years after surgery. The PCL in a PCLR prosthesis and the central tibial spine and femoral cam in a PCLS prosthesis might play comparable roles in determining the laxity in the posterior direction in these prostheses.     

A Parallel Domain Decomposition Algorithm for Simulating Blood Flow with Incompressible Navier-Stokes Equations with Resistive Boundary Condition

We introduce and study a parallel domain decomposition algorithm for the simulation of blood flow in compliant arteries using a fully-coupled system of nonlinear partial differential equations consisting of a linear elasticity equation and the incompressible Navier-Stokes equations with a resistive outflow boundary condition. The system is discretized with a finite element method on unstructured moving meshes and solved by a Newton-Krylov algorithm preconditioned with an overlapping restricted additive Schwarz method. The resistive outflow boundary condition plays an interesting role in the accuracy of the blood flow simulation and we provide a numerical comparison of its accuracy with the standard pressure type boundary condition. We also discuss the parallel performance of the implicit domain decomposition method for solving the fully coupled nonlinear system on a supercomputer with a few hundred processors.     

A Penalty Optimization Algorithm for Eigenmode Optimization Problem Using Sensitivity Analysis

This paper investigates the eigenmode optimization problem governed by the scalar Helmholtz equation in continuum system in which the computed eigenmode approaches the prescribed eigenmode in the whole domain. The first variation for the eigenmode optimization problem is evaluated by the quadratic penalty method, the adjoint variable method, and the formula based on sensitivity analysis. A penalty optimization algorithm is proposed, in which the density evolution is accomplished by introducing an artificial time term and solving an additional ordinary differential equation. The validity of the presented algorithm is confirmed by numerical results of the first and second eigenmode optimizations in 1D and 2D problems.     

A Continuum-Atomistic Multi-Timescale Algorithm for Micro/Nano Flows

A multi-timescale algorithm is proposed for simulating time-dependent problems in micro- and nano- fluidics. The total simulation domain is spatially decomposed into two regions. Molecular dynamics is employed in the crucial interfacial regions and continuum hydrodynamics is adopted in the remaining bulk regions. The coupling is through “constrained dynamics” in an overlap region. Our time scheme is based on the time scale separation between the continuum macro time step and molecular micro time step. This allows the molecular dynamics during one macro time step to be treated as in quasi-steady state. Therefore, molecular simulation is only performed in two shorter time intervals. Through linear extrapolation of macroscopic velocities and re-initialization of particle configurations, we can significantly reduce the total computational cost. We demonstrate and discuss our time algorithm through hybrid simulation of channel flow driven by a sinusoidally moving top wall. Converging results are achieved for cases of large separation of time scale with much less computational cost than with the original hybrid simulation without time extrapolation.     

Trigone sensitivity testing is a measure of exteroceptive function

The trigone sensitivity test, a complementary test to cystometry, has been proposed as a method for distinguishing certain clinical disorders. Pressure is applied to the trigone region of the bladder by pulling upon a Foley catheter with the balloon inflated and the amount of force needed to induce an urge to void is recorded. Although the trigone sensitivity test has been proposed as a test of exteroceptive function, it is possible that deeper receptors, perhaps proprioceptors, are actually responsible for the awareness of the urge to void during the study. The present investigation compared the trigone sensitivity test with provoked detrusor contraction in 107 patients. The findings indicate that variations in the two responses occur independently of each other. It is concluded that different neural pathways are responsible for the two functions. The hypothesis that exteroceptive and not proprioceptive nerves are being measured during trigone sensitivity testing is supported by these findings.     

Near-Field Imaging of Interior Cavities

A novel method is developed for solving the inverse problem of reconstructing the shape of an interior cavity. The boundary of the cavity is assumed to be a small and smooth perturbation of a circle. The incident field is generated by a point source inside the cavity. The scattering data is taken on a circle centered at the source. The method requires only a single incident wave at one frequency. Using a transformed field expansion, the original boundary value problem is reduced to a successive sequence of two-point boundary value problems and is solved in a closed form. By dropping higher order terms in the power series expansion, the inverse problem is linearized and an explicit relation is established between the Fourier coefficients of the cavity surface function and the total field. A nonlinear correction algorithm is devised to improve the accuracy of the reconstruction. Numerical results are presented to show the effectiveness of the method and its ability to obtain subwavelength resolution.     

Single lever Humphrey A.D.E. low flow universal anaesthetic breathing system. Part I: Comparison with dual lever A.D.E., Magill and Bain systems in anaesthetized spontaneously breathing adults

The single lever Humphrey A.D.E. anaesthetic system, in both coaxial and parallel (non-coaxial) forms, has recently been introduced. In principle the system offers efficient "universal" function by combining the advantages of Mapleson A, D and E systems. A within-patient comparison of its function in the Mapleson A mode (lever up) in spontaneously-breathing anaesthetized subjects was made to that of the original two lever A.D.E., the Magill (Mapleson A) and the Bain (Mapleson D) systems. The coaxial and parallel single lever A.D.E. systems functioned identically to each other and to the original two lever A.D.E. system, a mean fresh gas flow (FGF) of 51 ml X kg-1 X min-1 causing minimal rebreathing. Under identical conditions, the mean FGF required to just cause rebreathing increased to a mean of 71 ml X kg-1 X min-1 and 150 ml X kg-1 X min-1 with the Magill and the Bain systems respectively. With the single lever system, the switch to its Mapleson E mode for controlled ventilation involves the selection of the only alternative lever position (lever down) without further adjustment. The function and practical advantages in this E mode are presented in Part II.     

An Adaptive Perfectly Matched Layer Method for Multiple Cavity Scattering Problems

A uniaxial perfectly matched layer (PML) method is proposed for solving the scattering problem with multiple cavities. By virtue of the integral representation of the scattering field, we decompose the problem into a system of single-cavity scattering problems which are coupled with Dirichlet-to-Neumann maps. A PML is introduced to truncate the exterior domain of each cavity such that the computational domain does not intersect those for other cavities. Based on a posteriori error estimates, an adaptive finite element algorithm is proposed to solve the coupled system. The novelty of the proposed method is that its computational complexity is comparable to that for solving uncoupled single-cavity problems. Numerical experiments are presented to demonstrate the efficiency of the adaptive PML method.     

On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws

This paper is concerned with a new version of the Osher-Solomon Riemann solver and is based on a numerical integration of the path-dependent dissipation matrix. The resulting scheme is much simpler than the original one and is applicable to general hyperbolic conservation laws, while retaining the attractive features of the original solver: the method is entropy-satisfying, differentiable and complete in the sense that it attributes a different numerical viscosity to each characteristic field, in particular to the intermediate ones, since the full eigenstructure of the underlying hyperbolic system is used. To illustrate the potential of the proposed scheme we show applications to the following hyperbolic conservation laws: Euler equations of compressible gasdynamics with ideal gas and real gas equation of state, classical and relativistic MHD equations as well as the equations of nonlinear elasticity. To the knowledge of the authors, apart from the Euler equations with ideal gas, an Osher-type scheme has never been devised before for any of these complicated PDE systems. Since our new general Riemann solver can be directly used as a building block of high order finite volume and discontinuous Galerkin schemes we also show the extension to higher order of accuracy and multiple space dimensions in the new framework of P_NP_M schemes on unstructured meshes recently proposed in [9].     

Coronal laxity in extension in vivo after total knee arthroplasty

We performed stress arthrometric studies on 77 knees (71 patients) with total knee arthroplasty to determine the presence and magnitude of femoral abduction and adduction in knee extension. A total of 53 knees (49 patients) had posterior cruciate ligament-retaining (PCLR) prostheses, and 24 (22 patients) had PCL-substituting (PCLS) prostheses. The selected patients had successful arthroplasties with no clinical complications a minimum of 5 years after primary surgery. Each patient was subjected to a successive abduction and adduction stress test at 0°–20° of flexion using a Telos arthrometer. The mean values for abduction and adduction were 4.8° and 4.5° with a PCLR prosthesis, respectively, and 4.6° and 4.0° with a PCLS prosthesis. There were no statistical differences between PCLR and PCLS knees. The results suggest that approximately 4° of laxity in these directions is suitable in total knee arthroplasty for a satisfactory clinical outcome 5–9 years after surgery.     

Hybrid and Multiplicative Overlapping Schwarz Algorithms with Standard Coarse Spaces for Mixed Linear Elasticity and Stokes Problems

The goal of this work is to construct and study hybrid and multiplicative two-level overlapping Schwarz algorithms with standard coarse spaces for the almost incompressible linear elasticity and Stokes systems, discretized by mixed finite and spectral element methods with discontinuous pressures. Two different approaches are considered to solve the resulting saddle point systems: a) a preconditioned conjugate gradient (PCG) method applied to the symmetric positive definite reformulation of the almost incompressible linear elasticity system obtained by eliminating the pressure unknowns; b) a GMRES method with indefinite overlapping Schwarz preconditioner applied directly to the saddle point formulation of both the elasticity and Stokes systems. Condition number estimates and convergence properties of the proposed hybrid and multiplicative overlapping Schwarz algorithms are proven for the positive definite reformulation of almost incompressible elasticity. These results are based on our previous study [8] where only additive Schwarz preconditioners were considered for almost incompressible elasticity. Extensive numerical experiments with both finite and spectral elements show that the proposed overlapping Schwarz preconditioners are scalable, quasi-optimal in the number of unknowns across individual subdomains and robust with respect to discontinuities of the material parameters across subdomains interfaces. The results indicate that the proposed preconditioners retain a good performance also when the quasi-monotonicity assumption, required by the available theory, does not hold.     

A High Order Sharp-Interface Method with Local Time Stepping for Compressible Multiphase Flows

In this paper, a new sharp-interface approach to simulate compressible multiphase flows is proposed. The new scheme consists of a high order WENO finite volume scheme for solving the Euler equations coupled with a high order path-conservative discontinuous Galerkin finite element scheme to evolve an indicator function that tracks the material interface. At the interface our method applies ghost cells to compute the numerical flux, as the ghost fluid method. However, unlike the original ghost fluid scheme of Fedkiw et al. [15], the state of the ghost fluid is derived from an approximate-state Riemann solver, similar to the approach proposed in [25], but based on a much simpler formulation. Our formulation leads only to one single scalar nonlinear algebraic equation that has to be solved at the interface, instead of the system used in [25]. Away from the interface, we use the new general Osher-type flux recently proposed by Dumbser and Toro [13], which is a simple but complete Riemann solver, applicable to general hyperbolic conservation laws. The time integration is performed using a fully-discrete one-step scheme, based on the approaches recently proposed in [5, 7]. This allows us to evolve the system also with time-accurate local time stepping. Due to the sub-cell resolution and the subsequent more restrictive time-step constraint of the DG scheme, a local evolution for the indicator function is applied, which is matched with the finite volume scheme for the solution of the Euler equations that runs with a larger time step. The use of a locally optimal time step avoids the introduction of excessive numerical diffusion in the finite volume scheme. Two different fluids have been used, namely an ideal gas and a weakly compressible fluid modeled by the Tait equation. Several tests have been computed to assess the accuracy and the performance of the new high order scheme. A verification of our algorithm has been carefully carried out using exact solutions as well as a comparison with other numerical reference solutions. The material interface is resolved sharply and accurately without spurious oscillations in the pressure field.