SDF-Based ILW: Inverse Lax-Wendroff Method with the Signed Distance Function Representation of the Geometric Boundary

This paper studies the geometric boundary representations for Inverse Lax-Wendroff (ILW) method, aiming to develop a practical computer-aided engineering method without body-fitted meshes. We propose the signed distance function (SDF) representation of the geometric boundary and design an extremely efficient algorithm for foot point calculation, which is particularly in line with the needs of ILW. Theoretical and numerical analyses demonstrate that the SDF representation of geometric boundary can satisfy ILW’s needs better than others. The effectiveness and robustness of our proposed method are verified by simulating initial boundary value computational physical problems of Euler equation for compressible fluids.     

On Accurately Resolving Detonation Dynamics by Adaptive Finite Volume Method on Unstructured Grids

Long time simulations are needed in the numerical study of the Zeldovich-Neumann-Döring model, in which the quality resolving the dynamics of the detonation front is crucial. The numerical error introduced from the inappropriate outflow boundary condition and the mesh resolution are two main factors qualitatively affecting the dynamics of the detonation front. In this paper we improve the numerical framework in [15] by introducing the Strang splitting method and a new $h$-adaptive method with a feature based $a$ $posteriori$ error estimator. Then a cheap numerical approach is proposed to sharply estimate a time period, in which the unphysical influence on the detonation front can be avoided effectively. The sufficiently dense mesh resolution can be guaranteed around the detonation front and in the reaction zone by the proposed $h$-adaptive method. The numerical results show that the proposed method is sufficiently robust even for long time calculations, and the quality dynamics of the detonation can be obtained by the proposed numerical approach.     

A multi-mode shock tube for investigation of blast-induced traumatic brain injury

Blast-induced mild traumatic brain injury (bTBI) has become increasingly common in recent military conflicts. The mechanisms by which non-impact blast exposure results in bTBI are incompletely understood. Current small animal bTBI models predominantly utilize compressed air-driven membrane rupture as their blast wave source, while large animal models use chemical explosives. The pressure-time signature of each blast mode is unique, making it difficult to evaluate the contributions of the different components of the blast wave to bTBI when using a single blast source. We utilized a multi-mode shock tube, the McMillan blast device, capable of utilizing compressed air- and compressed helium-driven membrane rupture, and the explosives oxyhydrogen and cyclotrimethylenetrinitramine (RDX, the primary component of C-4 plastic explosives) as the driving source. At similar maximal blast overpressures, the positive pressure phase of compressed air-driven blasts was longer, and the positive impulse was greater, than those observed for shockwaves produced by other driving sources. Helium-driven shockwaves more closely resembled RDX blasts, but by displacing air created a hypoxic environment within the shock tube. Pressure-time traces from oxyhydrogen-driven shockwaves were very similar those produced by RDX, although they resulted in elevated carbon monoxide levels due to combustion of the polyethylene bag used to contain the gases within the shock tube prior to detonation. Rats exposed to compressed air-driven blasts had more pronounced vascular damage than those exposed to oxyhydrogen-driven blasts of the same peak overpressure, indicating that differences in blast wave characteristics other than peak overpressure may influence the extent of bTBI. Use of this multi-mode shock tube in small animal models will enable comparison of the extent of brain injury with the pressure-time signature produced using each blast mode, facilitating evaluation of the blast wave components contributing to bTBI.     

Adaptive Multi-Resolution Method for 3D Reactive Flows with Level Set Front Capturing

For compressible reactive flows with stiff source terms, a new block-based adaptive multi-resolution method coupled with the adaptive multi-resolution representation model for ZND detonation and a conservative front capturing method based on a level-set technique is presented. When simulating stiff reactive flows, underresolution in space and time can lead to incorrect propagation speeds of discontinuities, and numerical dissipation makes it impossible for traditional shock-capturing methods to locate the detonation front. To solve these challenges, the proposed method leverages an adaptive multi-resolution representation model to separate the scales of the reaction from those of fluid dynamics, achieving both high-resolution solutions and high efficiency. A level set technique is used to capture the detonation front sharply and reduce errors due to the inaccurate prediction of detonation speed. In order to ensure conservation, a conservative modified finite volume scheme is implemented, and the front transition fluxes are calculated by considering a Riemann problem. A series of numerical examples of stiff detonation simulations are performed to illustrate that the present method can acquire the correct propagation speed and accurately capture the sharp detonation front. Comparative numerical results also validate the approach’s benefits and excellent performance.     

Split Local Artificial Boundary Conditions for the Two-Dimensional Sine-Gordon Equation on R2

In this paper the numerical solution of the two-dimensional sine-Gordon equation is studied. Split local artificial boundary conditions are obtained by the operator splitting method. Then the original problem is reduced to an initial boundary value problem on a bounded computational domain, which can be solved by the finite difference method. Several numerical examples are provided to demonstrate the effectiveness and accuracy of the proposed method, and some interesting propagation and collision behaviors of the solitary wave solutions are observed.     

Wave Propagation Simulation Using the CIP Method of Characteristic Equations

We apply the CIP (Cubic Interpolated Profile) scheme to the numerical simulation of the acoustic wave propagation based on characteristic equations.The CIP scheme is based on a concept that both the wavefield and its spatial derivative propagate along the same characteristic curves derived from a hyperbolic differential equation. We describe the derivation of the characteristic equations for the acoustic waves from the basic equations by means of the directional splitting and the diagonalization of the coefficient matrix, and establish geophysical boundary conditions. Since the CIP scheme calculates both the wavefield and its spatial derivatives, it is easy to realize the boundary conditions theoretically. We also show some numerical simulation examples and the CIP can simulate acoustic wave propagation with high stability and less numerical dispersion. The method of characteristics with the CIP scheme is a very powerful technique to deal with the wave propagation in complex geophysical problems.     

Investigation of cavitation as a possible damage mechanism in blast-induced traumatic brain injury

Cavitation was investigated as a possible damage mechanism for war-related traumatic brain injury (TBI) due to an improvised explosive device (IED) blast. When a frontal blast wave encounters the head, a shock wave is transmitted through the skull, cerebrospinal fluid (CSF), and tissue, causing negative pressure at the contrecoup that may result in cavitation. Numerical simulations and shock tube experiments were conducted to determine the possibility of cranial cavitation from realistic IED non-impact blast loading. Simplified surrogate models of the head consisted of a transparent polycarbonate ellipsoid. The first series of tests in the 18-inch-diameter shock tube were conducted on an ellipsoid filled with degassed water to simulate CSF and tissue. In the second series, Sylgard gel, surrounded by a layer of degassed water, was used to represent the tissue and CSF, respectively. Simulated blast overpressure in the shock tube tests ranged from a nominal 10-25 pounds per square inch gauge (psig; 69-170?kPa). Pressure in the simulated CSF was determined by Kulite thin line pressure sensors at the coup, center, and contrecoup positions. Using video taken at 10,000 frames/sec, we verified the presence of cavitation bubbles at the contrecoup in both ellipsoid models. In all tests, cavitation at the contrecoup was observed to coincide temporally with periods of negative pressure. Collapse of the cavitation bubbles caused by the surrounding pressure and elastic rebound of the skull resulted in significant pressure spikes in the simulated CSF. Numerical simulations using the DYSMAS hydrocode to predict onset of cavitation and pressure spikes during cavity collapse were in good agreement with the tests. The numerical simulations and experiments indicate that skull deformation is a significant factor causing cavitation. These results suggest that cavitation may be a damage mechanism contributing to TBI that requires future study.     

An Application of the Level Set Method to Underwater Acoustic Propagation

An algorithm for computing wavefronts, based on the high frequency approximation to the wave equation, is presented. This technique applies the level set method to underwater acoustic wavefront propagation in the time domain. The level set method allows for computation of the acoustic phase function using established numerical techniques to solve a first order transport equation to a desired order of accuracy. Traditional methods for solving the eikonal equation directly on a fixed grid limit one to only the first arrivals, so these approaches are not useful when multi-path propagation is present. Applying the level set model to the problem allows for the time domain computation of the phase function on a fixed grid, without having to restrict to first arrival times. The implementation presented has no restrictions on range dependence or direction of travel, and offers improved efficiency over solving the full wave equation which under the high frequency assumption requires a large number of grid points to resolve the highly oscillatory solutions. Boundary conditions are discussed, and an approach is suggested for producing good results in the presence of boundary reflections. An efficient method to compute the amplitude from the level set method solutions is also presented. Comparisons to analytical solutions are presented where available, and numerical results are validated by comparing results with exact solutions where available, a full wave equation solver, and with wavefronts extracted from ray tracing software.     

Numerical Solution of Blow-Up Problems for Nonlinear Wave Equations on Unbounded Domains

The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered. Applying the unified approach, which is based on the operator splitting method, we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation, and reduce the nonlinear problem on the unbounded spatial domain to an initial-boundary-value problem on a bounded domain. Then the finite difference method is used to solve the reduced problem on the bounded computational domain. Finally, a broad range of numerical examples are given to demonstrate the effectiveness and accuracy of our method, and some interesting propagation and behaviors of the blow-up problems for nonlinear wave equations are observed.     

Principles of shock wave therapy

A shock wave is a transient pressure disturbance that propagates rapidly in three-dimensional space. It is associated with a sudden rise from ambient pressure to its maximum pressure. A significant tissue effect is cavitation consequent to the negative phase of the wave propagation. The current authors summarize the basic physics of shock waves and the physical parameters involved in assessing the amount of energy delivered to the target tissue and in comparing the various high- and low-energy devices being evaluated clinically for musculoskeletal applications.     

Upregulation of VEGF in Subchondral Bone of Necrotic Femoral Heads in Rabbits with Use of Extracorporeal Shock Waves

Extracorporeal shock wave treatment appears to be effective in patients with avascular necrosis of the femoral head. However, the pathway of biological events whereby this is accomplished has not been fully elucidated. The purpose of this study was to investigate the effect of extracorporeal shock waves on vascular endothelial growth factor (VEGF) expression in necrotic femoral heads of rabbits. VEGF expression was assessed by immunohistochemistry, quantitative real-time PCR, and Western blot analysis. The degree of angiogenesis was also assessed, as determined by the microvessel density (MVD), the assessment of which was based on CD31-expressing vessels. Bilateral avascular necrosis of femoral heads was induced with methylprednisolone and lipopolysaccharide in 30 New Zealand rabbits. The left limb (the study side) received shock wave therapy to the femoral head. The right limb (the control side) received no shock wave therapy. Biopsies of the femoral heads were performed at 1, 2, 4, 8, and 12 weeks. Western blot analysis and real-time PCR showed that shock wave therapy significantly increased VEGF protein and mRNA expression, respectively, in the subchondral bone of the treated necrotic femoral heads. Compared with the contralateral control without shock wave treatment, the VEGF mRNA expression levels increased to a peak at 2 weeks after the shock wave treatment and remained high for 8 weeks, then declined at 12 weeks, whereas the VEGF protein expression levels increased to a peak at 4 weeks after the shock wave treatment and remained high for 12 weeks. The immunostaining of VEGF was weak in the control group, and the immunoreactivity level in the shock-wave-treated group increased at 4 weeks and persisted for 12 weeks. The most intensive VEGF immunoreactivity was observed in the proliferative zone above the necrotic zone. At 4, 8, and 12 weeks after the shock wave treatment, MVD in subchondral bone from treated femoral heads was significantly higher than that in subchondral bone from untreated femoral heads. These data clearly show that extracorporeal shock waves can significantly upregulate the expression of VEGF. The upregulation of VEGF may play a role in inducing the ingrowth of neovascularization and in improving the blood supply to the femoral head.     

Development of a new biomechanical indicator for primary blast-induced brain injury

Primary blast-induced traumatic brain injury (bTBI) has been observed at the boundary of brain tissue and cerebrospinal fluid (CSF). Such injury can hardly be explained by using the theory of compressive wave propagation, since both the solid and fluid materials have similar compressibility and thus the intracranial pressure (ICP) has a continuous distribution across the boundary. Since they have completely different shear properties, it is hypothesized the injury at the interface is caused by shear wave. In the present study, a preliminary combined numerical and theoretical analysis was conducted based on the theory of shear wave propagation/reflection. Simulation results show that higher lateral acceleration of brain tissue particles is concentrated in the boundary region. Based on this finding, a new biomechanical vector, termed as strain gradient, was suggested for primary bTBI. The subsequent simple theoretical analysis reveals that this parameter is proportional to the value of lateral acceleration. At the boundary of lateral ventricles, high spatial strain gradient implies that the brain tissue in this area (where neuron cells may be contained) undergo significantly different strains and large velocity discontinuity, which may result in mechanical damage of the neuron cells.     

A High Frequency Boundary Element Method for Scattering by Convex Polygons with Impedance Boundary Conditions

We consider scattering of a time harmonic incident plane wave by a convex polygon with piecewise constant impedance boundary conditions. Standard finite or boundary element methods require the number of degrees of freedom to grow at least linearly with respect to the frequency of the incident wave in order to maintain accuracy. Extending earlier work by Chandler-Wilde and Langdon for the sound soft problem, we propose a novel Galerkin boundary element method, with the approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon. Theoretical analysis and numerical results suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency of the incident wave.     

Three-Dimensional Simulation of Balloon Dynamics by the Immersed Boundary Method Coupled to the Multiple-Relaxation-Time Lattice Boltzmann Method

The immersed boundary method (IBM) has been popular in simulating fluid structure interaction (FSI) problems involving flexible structures, and the recent introduction of the lattice Boltzmann method (LBM) into the IBM makes the method more versatile. In order to test the coupling characteristics of the IBM with the multiple-relaxation-time LBM (MRT-LBM), the three-dimensional (3D) balloon dynamics, including inflation, release and breach processes, are simulated. In this paper, some key issues in the coupling scheme, including the discretization of 3D boundary surfaces, the calculation of boundary force density, and the introduction of external force into the LBM, are described. The good volume conservation and pressure retention properties are verified by two 3D cases. Finally, the three FSI processes of a 3D balloon dynamics are simulated. The large boundary deformation and oscillation, obvious elastic wave propagation, sudden stress release at free edge, and recoil phenomena are all observed. It is evident that the coupling scheme of the IBM and MRT-LBM can handle complicated 3D FSI problems involving large deformation and large pressure gradients with very good accuracy and stability.     

High Order Finite Difference Methods with Subcell Resolution for Stiff Multispecies Discontinuity Capturing

In this paper, we extend the high order finite-difference method with subcell resolution (SR) in [34] for two-species stiff one-reaction models to multispecies and multireaction inviscid chemical reactive flows, which are significantly more difficult because of the multiple scales generated by different reactions. For reaction problems, when the reaction time scale is very small, the reaction zone scale is also small and the governing equations become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present SR method for reactive Euler system is a fractional step method. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with certain computed flow variables in the shock region modified by the Harten subcell resolution idea. Several numerical examples of multispecies and multireaction reactive flows are performed in both one and two dimensions. Studies demonstrate that the SR method can capture the correct propagation speed of discontinuities in very coarse meshes.     

Super-Grid Modeling of the Elastic Wave Equation in Semi-Bounded Domains

We develop a super-grid modeling technique for solving the elastic wave equation in semi-bounded two- and three-dimensional spatial domains. In this method, waves are slowed down and dissipated in sponge layers near the far-field boundaries. Mathematically, this is equivalent to a coordinate mapping that transforms a very large physical domain to a significantly smaller computational domain, where the elastic wave equation is solved numerically on a regular grid. To damp out waves that become poorly resolved because of the coordinate mapping, a high order artificial dissipation operator is added in layers near the boundaries of the computational domain. We prove by energy estimates that the super-grid modeling leads to a stable numerical method with decreasing energy, which is valid for heterogeneous material properties and a free surface boundary condition on one side of the domain. Our spatial discretization is based on a fourth order accurate finite difference method, which satisfies the principle of summation by parts. We show that the discrete energy estimate holds also when a centered finite difference stencil is combined with homogeneous Dirichlet conditions at several ghost points outside of the far-field boundaries. Therefore, the coefficients in the finite difference stencils need only be boundary modified near the free surface. This allows for improved computational efficiency and significant simplifications of the implementation of the proposed method in multi-dimensional domains. Numerical experiments in three space dimensions show that the modeling error from truncating the domain can be made very small by choosing a sufficiently wide super-grid damping layer. The numerical accuracy is first evaluated against analytical solutions of Lamb's problem, where fourth order accuracy is observed with a sixth order artificial dissipation. We then use successive grid refinements to study the numerical accuracy in the more complicated motion due to a point moment tensor source in a regularized layered material.     

Application of shock waves as a treatment modality in the vicinity of the brain and skull

OBJECT: Shock waves have not previously been used as a treatment modality for lesions in the brain and skull because of the lack of a suitable shock wave source and concerns about safety. Therefore, the authors have performed experiments aimed at developing both a new, compact shock wave generator with a holmium:yttrium-aluminum-garnet (Ho:YAG) laser and a safe method for exposing the surface of the brain to these shock waves. METHODS: Twenty male Sprague-Dawley rats were used in this study. In 10 rats, a single shock wave was delivered directly to the brain, whereas the protective effect of inserting a 0.7-mm-thick expanded polytetrafluoroethylene (ePTFE) dural substitute between the dura mater and skull before applying the shock wave was investigated in the other 10 rats. Visualizations on shadowgraphy along with pressure measurements were obtained to confirm that the shock wave generator was capable of conveying waves in a limited volume without harmful effects to the target. The attenuation rates of shock waves administered through a 0.7-mm-thick ePTFE dural substitute and a surgical cottonoid were measured to determine which of these materials was suitable for avoiding propagation of the shock wave beyond the target. CONCLUSIONS: Using the shock wave generator with the Ho:YAG laser, a localized shock wave (with a maximum overpressure of 50 bar) can be generated from a small device (external diameter 15 mm, weight 20 g). The placement of a 0.7-mm-thick ePTFE dural substitute over the dura mater reduces the overpressure of the shock wave by 96% and eliminates damage to surrounding tissue in the rat brain. These findings indicate possibilities for applying shock waves in various neurosurgical treatments such as cranioplasty, local drug delivery, embolysis, and pain management.     

Modified Upwinding Compact Scheme for Shock and Shock Boundary Layer Interaction

Standard compact scheme and upwinding compact scheme have high order accuracy and high resolution, but cannot capture the shock which is a discontinuity. This work developed a modified upwinding compact scheme which uses an effective shock detector to block compact scheme to cross the shock and a control function to mix the flux with WENO scheme near the shock. The new scheme makes the original compact scheme able to capture the shock sharply and, more importantly, keep high order accuracy and high resolution in the smooth area which is particularly important for shock boundary layer and shock acoustic interactions. Numerical results show the scheme is successful for 2-D Euler and 2-D Navier-Stokes solvers. The examples include 2-D incident shock, 2-D incident shock and boundary layer interaction. The scheme is robust, which does not involve case related parameters.     

A Multilevel Numerical Approach with Application in Time-Domain Electromagnetics

An algebraic multilevel method is proposed for efficiently simulating linear wave propagation using higher-order numerical schemes. This method is used in conjunction with the Finite Volume Time Domain (FVTD) technique for the numerical solution of the time-domain Maxwell's equations in electromagnetic scattering problems. In the multilevel method the solution is cycled through spatial operators of varying orders of accuracy, while maintaining highest-order accuracy at coarser approximation levels through the use of the relative truncation error as a forcing function. Higher-order spatial accuracy can be enforced using the multilevel method at a fraction of the computational cost incurred in a conventional higher-order implementation. The multilevel method is targeted towards electromagnetic scattering problems at large electrical sizes which usually require long simulation times due to the use of very fine meshes dictated by point-per-wavelength requirements to accurately model wave propagation over long distances.     

Enhanced kidney stone fragmentation by short delay tandem conventional and modified lithotriptor shock waves: a numerical analysis

PURPOSE: We evaluated the effectiveness of modified lithotriptor shock waves using computer models. MATERIALS AND METHODS: Finite element models were used to simulate the propagation of lithotriptor shock waves in human renal calculi in vivo. Kidney stones were assumed to be spherical, homogeneous, isotropic and linearly elastic, and immersed in a continuum fluid. Single and tandem shock wave pulses modified to intensify the collapse of cavitation bubbles near the stone surface to increase fragmentation efficiency and suppress the expansion of intraluminal bubbles for decreased vascular injury were analyzed. The effectiveness of the modified shock waves was assessed by comparing the states of loading in the renal calculi induced by these shock waves to those produced by conventional shock waves. RESULTS: Our numerical simulations revealed that modified shock waves produced marginally lower stresses in spherical renal calculi than those produced by conventional shock waves. Tandem pulses of conventional or modified shock waves produced peak stresses in the front and back halves of the renal calculi. However, the single shock wave pulses generated significant peak stresses in only the back halves of the renal calculi. CONCLUSIONS: Our numerical simulations suggest that for direct stress wave induced fragmentation modified shock waves should be as effective as conventional shock waves for fragmenting kidney stones. Also, with a small interval of 20 microseconds between the pulses tandem pulse lithotripsy using modified or conventional shock waves could be considerably more effective than single pulse lithotripsy for fragmenting kidney stones.