This document discusses dynamic absorbing boundary conditions for advective-diffusive systems with unknown Riemann invariants. It begins by motivating the need for absorbing boundary conditions to prevent non-convergent solutions and describes how a dissipative region can act as an absorbing layer. It then discusses boundary conditions for linear and nonlinear advective systems, presenting absorbing boundary conditions for 1D linear systems based on characteristic variables. For nonlinear problems, it proposes using the previous state as a reference state to linearize around.
Presentation at "Emerging problems in particle phenomenology" workshop held at CUNY on April 11, 2010. Has sensitivity of Jets+MET searches for 7 TeV LHC.
Presentation at "Emerging problems in particle phenomenology" workshop held at CUNY on April 11, 2010. Has sensitivity of Jets+MET searches for 7 TeV LHC.
Talk given at the Particle Technology Lab, Zurich, Switzerland, November 2008.larry77
The process of nanoparticle agglomeration as a function of the monomer-monomer interaction potential is simulated numerically by solving Langevin equations for a set of interacting monomers in three dimensions. The simulation results are used to investigate the structure of the generated clusters and the collision frequency between small clusters. Cluster restructuring is also observed and discussed. We identify a time-dependent fractal dimension whose evolution is linked to the kinetics of two cluster populations. The absence of screening in the Langevin equations is discussed and its effect on cluster translational and rotational properties is quantified.
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALLKees Nieuwstad
The motion induced by gravity of solid spheres in a vessel filled with fluid has been investigated experimentally at Reynolds numbers in the range from 1-74 and Stokes numbers ranging from 0.2-17. Trajectories of the spheres have been measured with a focus on start-up behavior, and on impact with a horizontal wall. Two models have been investigated. The first describes the accelerating motion of the sphere. The second model predicts the distance from the wall at which the sphere starts decelerating.
The flow in the vicinity of the sphere was measured by means of PIV. The time scales and flow structures strongly depend on the Reynolds number. Measurements performed are in good agreement with simulations performed at the Kramers Laboratorium.
Advances in the Solution of NS Eqs. in GPGPU Hardware. Second order scheme an...Storti Mario
A Navier-Stokes solver based on Cartesian structured finite volume
discretization with embedded bodies is presented. Fluid structure
interaction with solid bodies is performed with an explicit
partitioned strategy. The Navier-Stokes equations are solved in the
whole domain via a Semi-Implicit Method for Pressure Linked Equations
(SIMPLE) using a colocated finite volume scheme, stabilized
via the Rhie-Chow discretization. As uniform Cartesian grids are used,
the solid interface usually do not coincide with the mesh, and then a
second order Immersed Boundary Method is proposed, in order to avoid
the loss of precision due to the staircase representation of the
surface. This fact also affects the computation of fluid forces on the
solid wall and, accordingly, the results in the fluid-structure
analysis. In the present work, first and second order approximations
for computing the fluid forces at the interface are studied and
compared. The solver is specially oriented to General Purpose Graphic
Processing Units (GPGPU) hardware and the efficiency is
discussed. Moreover, a novel submerged buoy experiment is also
reported and serves to validate the presented fluid-structure
algorithm. The experiment consists of a sphere with positive buoyancy
fully submerged in a cubic tank, subject to harmonic displacements
imposed by a shake table. The sphere is attached to the bottom of
the tank with a string. Position of the buoy is determined from video
records with a Motion Capture algorithm. The obtained amplitude and
phase curves allow a precise determination of the added mass and drag
forces. Due to this aspect the experimental data can be of interest
for the validation of fluid-structure interaction codes. Finally, the
numerical results are compared with the experiments, and allows the
validation of the numerically predicted drag and added mass of the
body.
Comparison of Different Absorbing Boundary Conditions for GPR Simulation by t...IJMER
This paper compares three boundary conditions, i.e. transmitting boundary condition, Sarma
absorbing boundary condition and the uniaxial complete matched layerabsorbing boundary condition for
simulation of ground penetrating radar (GPR) by the time domain finite element (FEM) method. The
formulations of the three boundary conditions for the FEM method are described. Their effectiveness in
absorbing the incident electromagnetic waves are evaluated by the reflection coefficient on the boundary
of a simple GPR model.The results demonstrate that UPML boundary condition can yield a reflection
coefficient smaller than -50 dB, which is -20 dB smaller than other two boundary conditions.
Hybrid Engine (Stirling Engine + IC Engine + Electric Motor)IJMER
Hybrid engine is a combination of Stirling engine, IC engine and Electric motor. All these 3 are
connected together to a single shaft. The power source of the Stirling engine will be a Solar Panel. The aim of
this is to run the automobile using a Hybrid engine
Talk given at the Particle Technology Lab, Zurich, Switzerland, November 2008.larry77
The process of nanoparticle agglomeration as a function of the monomer-monomer interaction potential is simulated numerically by solving Langevin equations for a set of interacting monomers in three dimensions. The simulation results are used to investigate the structure of the generated clusters and the collision frequency between small clusters. Cluster restructuring is also observed and discussed. We identify a time-dependent fractal dimension whose evolution is linked to the kinetics of two cluster populations. The absence of screening in the Langevin equations is discussed and its effect on cluster translational and rotational properties is quantified.
PIV EXPERIMENTS ON THE FLOW INDUCED BY A SPHERE SEDIMENTING TOWARDS A SOLID WALLKees Nieuwstad
The motion induced by gravity of solid spheres in a vessel filled with fluid has been investigated experimentally at Reynolds numbers in the range from 1-74 and Stokes numbers ranging from 0.2-17. Trajectories of the spheres have been measured with a focus on start-up behavior, and on impact with a horizontal wall. Two models have been investigated. The first describes the accelerating motion of the sphere. The second model predicts the distance from the wall at which the sphere starts decelerating.
The flow in the vicinity of the sphere was measured by means of PIV. The time scales and flow structures strongly depend on the Reynolds number. Measurements performed are in good agreement with simulations performed at the Kramers Laboratorium.
Advances in the Solution of NS Eqs. in GPGPU Hardware. Second order scheme an...Storti Mario
A Navier-Stokes solver based on Cartesian structured finite volume
discretization with embedded bodies is presented. Fluid structure
interaction with solid bodies is performed with an explicit
partitioned strategy. The Navier-Stokes equations are solved in the
whole domain via a Semi-Implicit Method for Pressure Linked Equations
(SIMPLE) using a colocated finite volume scheme, stabilized
via the Rhie-Chow discretization. As uniform Cartesian grids are used,
the solid interface usually do not coincide with the mesh, and then a
second order Immersed Boundary Method is proposed, in order to avoid
the loss of precision due to the staircase representation of the
surface. This fact also affects the computation of fluid forces on the
solid wall and, accordingly, the results in the fluid-structure
analysis. In the present work, first and second order approximations
for computing the fluid forces at the interface are studied and
compared. The solver is specially oriented to General Purpose Graphic
Processing Units (GPGPU) hardware and the efficiency is
discussed. Moreover, a novel submerged buoy experiment is also
reported and serves to validate the presented fluid-structure
algorithm. The experiment consists of a sphere with positive buoyancy
fully submerged in a cubic tank, subject to harmonic displacements
imposed by a shake table. The sphere is attached to the bottom of
the tank with a string. Position of the buoy is determined from video
records with a Motion Capture algorithm. The obtained amplitude and
phase curves allow a precise determination of the added mass and drag
forces. Due to this aspect the experimental data can be of interest
for the validation of fluid-structure interaction codes. Finally, the
numerical results are compared with the experiments, and allows the
validation of the numerically predicted drag and added mass of the
body.
Comparison of Different Absorbing Boundary Conditions for GPR Simulation by t...IJMER
This paper compares three boundary conditions, i.e. transmitting boundary condition, Sarma
absorbing boundary condition and the uniaxial complete matched layerabsorbing boundary condition for
simulation of ground penetrating radar (GPR) by the time domain finite element (FEM) method. The
formulations of the three boundary conditions for the FEM method are described. Their effectiveness in
absorbing the incident electromagnetic waves are evaluated by the reflection coefficient on the boundary
of a simple GPR model.The results demonstrate that UPML boundary condition can yield a reflection
coefficient smaller than -50 dB, which is -20 dB smaller than other two boundary conditions.
Hybrid Engine (Stirling Engine + IC Engine + Electric Motor)IJMER
Hybrid engine is a combination of Stirling engine, IC engine and Electric motor. All these 3 are
connected together to a single shaft. The power source of the Stirling engine will be a Solar Panel. The aim of
this is to run the automobile using a Hybrid engine
An air pulse generating fruit harvesting device that causes the oscillation and detaching of said fruits by fatigue and breakage of the stems of the same, that comprises at least two rotating elements: a first rotating element that generates an air current and a second rotating element that movably guides said pulses of air; and also comprises static blades that guide said air current. A process to harvest fruits with said device.
Advances in the Solution of Navier-Stokes Eqs. in GPGPU Hardware. Modelling F...Storti Mario
In this article we compare the results obtained with an implementation of the Finite Volume for structured meshes on GPGPUs with experimental results and also with a Finite Element code with boundary fitted strategy. The example is a fully submerged spherical buoy immersed in a cubic water recipient. The recipient undergoes an harmonic linear motion imposed with a shake table. The experiment is recorded with a high speed camera and the displacement of the buoy if obtained from the video with a MoCap (Motion Capture) algorithm. The amplitude and phase of the resulting motion allows to determine indirectly the added mass and drag of the sphere.
Optimization of Technological Process to Decrease Dimensions of Circuits XOR,...ijfcstjournal
The paper describes an approach of increasing of integration rate of elements of integrated circuits. The
approach has been illustrated by example of manufacturing of a circuit XOR. Framework the approach one
should manufacture a heterostructure with specific configuration. After that several special areas of the
heterostructure should be doped by diffusion and/or ion implantation and optimization of annealing of dopant and/or radiation defects. We analyzed redistribution of dopant with account redistribution of radiation
defects to formulate recommendations to decrease dimensions of integrated circuits by using analytical
approaches of modeling of technological process.
AN APPROACH TO OPTIMIZE OF MANUFACTURING OF A VOLTAGE REFERENCE BASED ON HETE...JaresJournal
In this paper we introduce an approach to increase density of field-effect transistors framework a voltage
reference. Framework the approach we consider manufacturing the inverter in heterostructure with specific configuration. Several required areas of the heterostructure should be doped by diffusion or ion implantation. After that dopant and radiation defects should by annealed framework optimized scheme. We also
consider an approach to decrease value of mismatch-induced stress in the considered heterostructure. We
introduce an analytical approach to analyze mass and heat transport in heterostructures during manufacturing of integrated circuits with account mismatch-induced stress.
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
ON INCREASING OF DENSITY OF ELEMENTS IN A MULTIVIBRATOR ON BIPOLAR TRANSISTORSijcsitcejournal
In this paper we consider an approach to increase density of elements of a multivibrator on bipolar transistors.
The considered approach based on manufacturing a heterostructure with necessity configuration,
doping by diffusion or ion implantation of required areas to manufacture the required type of conductivity
(p or n) in the areas and optimization of annealing of dopant and/or radiation defects to manufacture more
compact distributions of concentrations of dopants. We also introduce an analytical approach to prognosis
technological process.
ON OPTIMIZATION OF MANUFACTURING OF FIELD-EFFECT HETEROTRANSISTORS FRAMEWORK ...ijoejournal
In this paper we introduce an approach to increase density of field-effect transistors framework a voltage reference. Framework the approach we consider manufacturing the inverter in heterostructure with specific configuration. Several required areas of the heterostructure should be doped by diffusion or ion implantation. After that dopant and radiation defects should by annealed framework optimized scheme. We also consider an approach to decrease value of mismatch-induced stress in the considered heterostructure. We introduce an analytical approach to analyze mass and heat transport in heterostructures during manufacturing of integrated circuits with account mismatch-induced stress.
AN APPROACH TO OPTIMIZE MANUFACTURE OF AN ACTIVE QUADRATURE SIGNAL GENERATOR ...antjjournal
In this paper we introduce an approach to increase density of field-effect transistors framework an active
quadrature signal generator. Framework the approach we consider manufacturing the generator in heterostructure
with specific configuration. Several required areas of the heterostructure should be doped by diffusion
or ion implantation. After that dopant and radiation defects should by annealed framework optimized
scheme. We also consider an approach to decrease value of mismatch-induced stress in the considered
heterostructure. We introduce an analytical approach to analyze mass and heat transport in heterostructures
during manufacturing of integrated circuits with account mismatch-induced stress.
Accuracy of the internal multiple prediction when a time-saving method based ...Arthur Weglein
The inverse scattering series (ISS) is a direct inversion method for a multidimensional acoustic,
elastic and anelastic earth. It communicates that all inversion processing goals can be
achieved directly and without any subsurface information. This task is reached through a taskspecific
subseries of the ISS. Using primaries in the data as subevents of the first-order internal
multiples, the leading-order attenuator can predict the time of all the first-order internal multiples
and is able to attenuate them.
Optimization of technological process to decrease dimensions of circuits xor ...ijfcstjournal
The paper describes an approach of increasing of integration rate of elements of integrated circuits. The
approach has been illustrated by example of manufacturing of a circuit XOR. Framework the approach one
should manufacture a heterostructure with specific configuration. After that several special areas of the
heterostructure should be doped by diffusion and/or ion implantation and optimization of annealing of dopant
and/or radiation defects. We analyzed redistribution of dopant with account redistribution of radiation
defects to formulate recommendations to decrease dimensions of integrated circuits by using analytical
approaches of modeling of technological process.
The resolution of Computational Fluid Dynamics (CFD) problems on Graphic Processing
Units (GPU’s) requires of specialized algorithms due to the particular hardware architecture of these
devices. Algorithms that fall in the category of cellular automata (CA) are the best fitted, for instance
explicit Finite Volume or Finite Element methods. But in the case of incompressible flow it is not possible
to develop a pure explicit algorithm, due to the essentially non-local character of the incompressibility
condition. In this case the algorithms that are closer to an explicit approach, are segregated algorithms,
like the Fractional Step Method. In these algorithms the more time consuming stage is (asymptotically
for large problems) the solution of the Poisson’s equation for pressure. A common choice for it’s solution
is the IOP (Iterated Orthogonal Projection) method, which requires a series of solutions on the complete
mesh. In this work a variant of the IOP, called Accelerated Global Preconditioning (AGP), is proposed.
It is based on using a Preconditioned Conjugate Gradient (which is an accelerated iterative method, in
contrast with the stationary scheme used in IOP ) for the pressure on the fluid, and preconditioning with
the solution on the global domain (fluid and solid). Of course, solving the problem on the global domain
represents more computational work than solving the problem only in the fluid, but this can be faster
in a structured mesh if a fast solvers as Multigrid or Fast Fourier Transform (FFT) is used. The main
advantage of AGP over IOP is that it is an accelerated solver, whereas the IOP is stationary. In addition
AGP iterates only on pressure, whereas IOP iterates on both pressure and velocity.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024Neo4j
Neha Bajwa, Vice President of Product Marketing, Neo4j
Join us as we explore breakthrough innovations enabled by interconnected data and AI. Discover firsthand how organizations use relationships in data to uncover contextual insights and solve our most pressing challenges – from optimizing supply chains, detecting fraud, and improving customer experiences to accelerating drug discoveries.
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
available on those devices, but many of the features provide convenience and capability but sacrifice security. This best practices guide outlines steps the users can take to better protect personal devices and information.
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
UiPath Test Automation using UiPath Test Suite series, part 5DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 5. In this session, we will cover CI/CD with devops.
Topics covered:
CI/CD with in UiPath
End-to-end overview of CI/CD pipeline with Azure devops
Speaker:
Lyndsey Byblow, Test Suite Sales Engineer @ UiPath, Inc.
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
A tale of scale & speed: How the US Navy is enabling software delivery from l...sonjaschweigert1
Rapid and secure feature delivery is a goal across every application team and every branch of the DoD. The Navy’s DevSecOps platform, Party Barge, has achieved:
- Reduction in onboarding time from 5 weeks to 1 day
- Improved developer experience and productivity through actionable findings and reduction of false positives
- Maintenance of superior security standards and inherent policy enforcement with Authorization to Operate (ATO)
Development teams can ship efficiently and ensure applications are cyber ready for Navy Authorizing Officials (AOs). In this webinar, Sigma Defense and Anchore will give attendees a look behind the scenes and demo secure pipeline automation and security artifacts that speed up application ATO and time to production.
We will cover:
- How to remove silos in DevSecOps
- How to build efficient development pipeline roles and component templates
- How to deliver security artifacts that matter for ATO’s (SBOMs, vulnerability reports, and policy evidence)
- How to streamline operations with automated policy checks on container images
Sudheer Mechineni, Head of Application Frameworks, Standard Chartered Bank
Discover how Standard Chartered Bank harnessed the power of Neo4j to transform complex data access challenges into a dynamic, scalable graph database solution. This keynote will cover their journey from initial adoption to deploying a fully automated, enterprise-grade causal cluster, highlighting key strategies for modelling organisational changes and ensuring robust disaster recovery. Learn how these innovations have not only enhanced Standard Chartered Bank’s data infrastructure but also positioned them as pioneers in the banking sector’s adoption of graph technology.
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
SAP Sapphire 2024 - ASUG301 building better apps with SAP Fiori.pdfPeter Spielvogel
Building better applications for business users with SAP Fiori.
• What is SAP Fiori and why it matters to you
• How a better user experience drives measurable business benefits
• How to get started with SAP Fiori today
• How SAP Fiori elements accelerates application development
• How SAP Build Code includes SAP Fiori tools and other generative artificial intelligence capabilities
• How SAP Fiori paves the way for using AI in SAP apps
SAP Sapphire 2024 - ASUG301 building better apps with SAP Fiori.pdf
Dynamic Absorbing Boundary Conditions
1. Dynamic absorbing boundary conditions by M.Storti et.al.
Dynamic Absorbing Boundary Conditions for
Advective-Difusive Systems with Unknown
Riemann Invariants
by Mario Storti, Rodrigo Paz, Luciano Garelli, Lisandro Dalc´n
ı
´
Centro Internacional de Metodos Computacionales
en Ingenier´a - CIMEC
ı
INTEC, (CONICET-UNL), Santa Fe, Argentina
<mario.storti at gmail.com>
http://www.cimec.org.ar/mstorti
´
Centro Internacional de Metodos Computacionales en Ingenier´a
ı 1
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2. Dynamic absorbing boundary conditions by M.Storti et.al.
Motivation for Absorbing Boundary Conditions
In wave-like propagation problems, not
including ABC may lead to
non-convergent solutions .
utt = c2 ∆u, in Ω
u = u(x, t), at Γb , u = 0, at Γ∞ v
¯
u doesn’t converge to the correct
solution irradiating energy from the
source, even if Γ∞ → ∞. A standing
wave is always found.
u is unbounded if u emits in an
¯
eigenfrequency which is a resonance
mode of the closed cavity.
Absorbing boundary conditions must be
added to the outer boundary in order to let
energy be extracted from the domain.
´
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3. Dynamic absorbing boundary conditions by M.Storti et.al.
Motivation for Absorbing Boundary Conditions (cont.)
If a dissipative region is large
enough so that many wavelengths
are included in the region may act
as an absorbing layer, i.e. as an
absorbing boundary condition.
dissipative
However, it is a common
misconception to assume that a
coarse mesh adds dissipation and
fine mesh coarse mesh
consequently may improve
absorption. For instance for the dissipative??
wave equation a coarse mesh may
lead to evanescent solutions and
then to act as a fully reflecting
boundary.
´
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4. Dynamic absorbing boundary conditions by M.Storti et.al.
Boundary conditions for advective diffusive systems
Well known theory and practice for advective systems say that at a boundary
the number of Dirichlet conditions should be equal to the
number of incoming characteristics .
∂U ∂Fc,j (U)
+ =0
∂t ∂xj
∂Fc,j (U)
Ac,j = , advective Jacobian
∂U
Nbr. of incoming characteristics = sum(eig(A · n) < 0)
ˆ
ˆ
n is the exterior normal.
Adding extra Dirichlet conditions leads to spurious shocks, and lack of
enough Dirichlet conditions leads to instability.
´
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5. Dynamic absorbing boundary conditions by M.Storti et.al.
Boundary conditions for advective diffusive systems (cont.)
For simple scalar advection problems the Jacobian is the transport velocity.
The rule is then to check the projection of velocity onto the exterior normal.
For more complex flows (i.e. with non diagonalizable Jacobians , as gas
dynamics or shallow water eqs.) the number of incoming characteristics may
be approx. predicted from the flow conditions.
subsonic flow (Minf<1) supersonic flow (Minf>1)
supersonic outgoing
subsonic incoming (no fields imposed)
rho,u,v rho,u,v,p
ck
sho
M>1 b ow
M<1
1111111
0000000
1111111
0000000 111111
000000
111111
000000
1111111
0000000 111111
000000
1111111
0000000 111111
000000
p p
subsonic outgoing subsonic outgoing
´
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6. Dynamic absorbing boundary conditions by M.Storti et.al.
Absorbing boundary conditions
However, this kind of conditions are, generally, reflective . Consider a pure
advective system of equations in 1D, i.e., Fd,j ≡ 0
∂H(U) ∂Fc,x (U)
+ = 0, in [0, L]. (1)
∂t ∂x
If the system is “linear”, i.e., Fc,x (U) = AU, H(U) = CU (A and C do
not depend on U), a first order linear system is obtained
∂U ∂U
C +A = 0. (2)
∂t ∂x
The system is “hyperbolic” if C is invertible, C−1 A is diagonalizable with
real eigenvalues. If this is the case, it is possible to make the following
eigenvalue decomposition for C−1 A
C−1 A = SΛS−1 , (3)
where S is real and invertible and Λ is real and diagonal. If new variables are
´
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7. Dynamic absorbing boundary conditions by M.Storti et.al.
defined V = S−1 U, then equation (2) becomes
∂V ∂V
+Λ = 0. (4)
∂t ∂x
Now, each equation is a linear scalar advection equation
∂vk ∂vk
+ λk = 0, (no summation over k ). (5)
∂t ∂x
vk are the “characteristic components” and λk are the “characteristic
velocities” of propagation.
´
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8. Dynamic absorbing boundary conditions by M.Storti et.al.
Linear 1D absorbing boundary conditions
Assuming λk = 0, the absorbing boundary conditions are, depending on the
sign of λk ,
if λk > 0: vk (0) = vk0 ;
¯ no boundary condition at x =L
(6)
if λk < 0: vk (L) = vkL ; no boundary condition at x = 0
¯
This can be put in compact form as
¯
Π+ (V − V0 ) = 0; at x =0
V
(7)
Π− (V
V
¯
− VL ) = 0; at x =L
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9. Dynamic absorbing boundary conditions by M.Storti et.al.
Linear 1D absorbing boundary conditions (cont.)
Π± are the projection matrices onto the right/left-going characteristic modes
V
in the V basis,
1; if j = k and λ > 0
k
Π+ =
V,jk
0; otherwise, (8)
Π+ + Π− = I.
It can be easily shown that they are effectively projection matrices, i.e.,
Π± Π± = Π± and Π+ Π− = 0. Coming back to the boundary condition at
x = L in the U basis, it can be written
Π− S−1 (U − UL ) = 0
V
¯ (9)
or, multiplying by S at the left
Π± (U − U0,L ) = 0, at x = 0, L,
U
¯ (10)
where
Π± = S Π± S−1 ,
U V (11)
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10. Dynamic absorbing boundary conditions by M.Storti et.al.
Linear 1D absorbing boundary conditions (cont.)
Π± (U − U0,L ) = 0, at x = 0, L,
U
¯
(12)
Π±
U = S Π±
V
−1
S ,
These conditions are completely absorbing for 1D linear advection system of
equations (2).
+
The rank of Π is equal to the number n+ of positive eigenvalues, i.e., the
number of right-going waves. Recall that the right-going waves are incoming
at the x = 0 boundary and outgoing at the x = L boundary. Conversely, the
−
rank of Π is equal to the number n− of negative eigenvalues, i.e., the
number of left-going waves (incoming at x = L and outgoing at the x = 0
boundary).
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11. Dynamic absorbing boundary conditions by M.Storti et.al.
ABC for nonlinear problems
First order absorbing boundary conditions may be constructed by imposing
exactly the components along the incoming characteristics.
Π− (Uref ) (U − Uref ) = 0.
Π− is the projection operator onto incoming characteristics. It can be
obtained straightforwardly from the projected Jacobian.
This assumes linearization of the equations around a state Uref . For linear
problems Ac,j do not depend on U, and then neither the projection operator,
so that absorbing boundary conditions coefficients are constant.
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12. Dynamic absorbing boundary conditions by M.Storti et.al.
ABC for nonlinear problems (cont.)
For non-linear problems the Jacobian and projection operator may vary and
then the above mentioned b.c.’s are not fully absorbing.
In some cases the concept of characteristic component may be extended to
the non-linear case: the “Riemann invariants” . Fully absorbing boundary
conditions could be written in terms of the invariants:
wj = wref,j , if wj is an incoming R.I.
R.I. are computed analytically. There are no automatic (numerical)
techniques to compute them. (They amount to compute an integral in
phase space along a specific path ).
R.I. are known for shallow water, channel flow (for rectangular
√ √
wj = u · n ± 2 gh, and triangular channel shape wj = u · n ± 4 gh).
ˆ ˆ
For gas dynamics the well known R.I. in fact are invariants only under
isentropic conditions (i.e. not truly invariant).
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13. Dynamic absorbing boundary conditions by M.Storti et.al.
ABC for nonlinear problems (cont.)
Search for an absorbing boundary condition that
should be fully absorbent in non-linear conditions, and
can be computed numerically (no need of analytic expressions like R.I.)
Solution: Use last state as reference state, ULSAR .
Uref = Un , n = time step number.
Π− (Un ) (Un+1 − Un ) = 0.
As Un+1 − Un is usually small, linearization is valid.
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14. Dynamic absorbing boundary conditions by M.Storti et.al.
ABC for nonlinear problems (cont.)
Disadvantage: Flow conditions are only determined from the initial state. No
external information comes from the outside.
Solution: use a combination of linear/R.I. b.c.’s on incoming boundaries and
use fully non-linear a.b.c’s with previous state as reference state at the outlet.
subsonic/supersonic flow
n
Uref=Uinf Uref=U
1111111
0000000
1111111
0000000
1111111
0000000
1111111
0000000
1111111
0000000
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15. Dynamic absorbing boundary conditions by M.Storti et.al.
Dynamic boundary conditions
As the flow is computed it may happen that the number of characteristics
changes in time. Two examples follow.
Think at transport of a scalar on top of a velocity field obtained by an
incompressible Navier-Stokes solver (not truly that in the example since both
systems are coupled). If the exterior flow is not modeled then flow at the top
opening may be reverted.
ventilacion
perfil de viento
inyeccion de carbon
ventilacion ventilacion
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16. Dynamic absorbing boundary conditions by M.Storti et.al.
Dynamic boundary conditions (cont.)
If the interior is modeled only, then it’s natural to leave concentration free at
the top opening. However the flow can revert at some portions of the opening
producing a large incoming of undetermined values (in practice large negative
concentrations are observed). Imposing a value at the opening is stable but
would lead to a spurious discontinuity at the outlet.
The ideal would be to switch dynamically from one condition to the other
during the computation.
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17. Dynamic absorbing boundary conditions by M.Storti et.al.
Nozzle chamber fill
The case is the ignition of a rocket launcher nozzle in a low pressure
◦
atmosphere. The fluid is initially at rest (143 Pa, 262 K). At time t = 0 a
membrane at the throat is broken. Behind the membrane there is a reservoir at
◦
5
6×10 Pa, 4170 K. A strong shock (intensity p1 /p2 >1000) propagates
from the throat to the outlet. The gas is assumed as ideal (γ = 1.17). In the
steady state a supersonic flow with a max. Mach of 4 at the outlet is found.
The objective of the simulation is to determine the time needed to fill the
chamber (< 1msec) and the final steady flow.
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18. Dynamic absorbing boundary conditions by M.Storti et.al.
Nozzle chamber fill (cont.)
We impose density, pressure and tangential velocity at inlet (assuming
subsonic inlet), slip condition at the nozzle wall. The problem is with the outlet
boundary. Initially the flow is subsonic (fluid at rest) there, and switches to
supersonic. The rule dictaminates to impose 1 condition, as a subsonic outlet
(may be pressure, which is known) and no conditions after (supersonic
outlet). If pressure is imposed during the wall computation, then a spurious
shock is formed at the outlet.
This test case has been contrasted with experimental data obtained at
ESTEC/ESA (European Space Research and Technology Centre-European
Space Agency, Noordwijk, Holanda). The predicted mean velocity was
2621 m/s to be compared with the experimental value of 2650 ± 50 m/sec.
Again, the ideal would be to switch dynamically from one condition to the
other during the computation .
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19. Dynamic absorbing boundary conditions by M.Storti et.al.
Object falling at supersonic speed
11
00
11
00
11
00
11
00
11
00
11
00
Consider, for simplicity, a two 11
00
11
00
11
00
dimensional case of an
homogeneous ellipse in free fall. As A
the body accelerates, the pitching 11111
00000
11111
00000
11111
00000
11111
00000
moments tend to increase the angle
of attack until it stalls (A), and then 111
000
111
000
111
000
the body starts to fall towards its 111
000B
111
000
111
000
other end and accelerating etc... 111
000 C
111
000
11111
00000
( “flutter” ). However, if the body 11111
00000
11111
00000
11111
00000
has a large angular moment at (B)
111
000 111
000
then it may happen that it rolls on 111
000 111
000
111
000 111
000
111
000 111
000
itself, keeping always the same 111
000 111
000
111
000 111
000
111
000 111
000
111
000
sense of rotation. This kind of falling 111
000
11111
00000 1111
0000
11111
00000 1111
0000
mechanism is called “tumbling” 11111
00000 1111
0000
11111
00000 1111
0000
and is characteristic of less slender
and more massive objects. flutter tumble
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20. Dynamic absorbing boundary conditions by M.Storti et.al.
Object falling at supersonic speed (cont.)
Under certain conditions in size and density relation to the surrounding
atmosphere it reaches supersonic speeds. In particular as form drag grows
like L2 whereas weight grows like L3 , larger bodies tend to reach larger limit
speeds and eventually reach supersonic regime. At supersonic speeds the
principal source of drag is the shock wave, we use slip boundary condition at
the body in order to simplify the problem.
We also do the computation in a 11
00
11
00
non-inertial system following the body,
11
00
11
00
11
00
11
00
so that non-inertial terms (Coriolis,
11
00
11
00
centrifugal, etc...) are added. In this 11
00
frame some portions of the boundary
are alternatively in all the conditions
(subsonic incoming, subsonic 11111
00000
11111
00000
outgoing, supersonic incoming, 11111
00000
supersonic outgoing). 11111
00000
11111
00000
Again, the ideal would be to switch 11111
00000 11111
00000
dynamically from one condition to 11111
00000
11111
00000
the other during the computation.
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21. Dynamic absorbing boundary conditions by M.Storti et.al.
Object falling at supersonic speed (cont.)
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22. Dynamic absorbing boundary conditions by M.Storti et.al.
Object falling at supersonic speed (cont.)
Whether RI or ULSAR based boundary conditions are used, if the number of
incoming/outgoing characteristics vary in time this requires to change the
profile of the system matrix during time evolution. In order to do this we add
dynamic boundary conditions either via Lagrange multipliers or penalization.
However this techniques add extra bad conditioning to the system of
equations so that special iterative methods are needed.
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23. Dynamic absorbing boundary conditions by M.Storti et.al.
Object falling at supersonic speed (cont.)
∂U ∂U
C +A = 0.
∂t ∂x
Consider for simplicity a linear system of advective equations discretized with
centered (i.e. no upwind) finite differences
Un+1 − Un
0 0 Un+1 − Un+1
C +A 1 0
= 0;
∆t h
Un+1 − Un Un+1 − Un+1
C k k
+ A k+1 k−1
= 0, k ≥ 1
∆t 2h
k ≥ 0 node index, n ≥ 0 time index, h = mesh size, C, A =enthalpy and
advective Jacobians.
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24. Dynamic absorbing boundary conditions by M.Storti et.al.
Object falling at supersonic speed (cont.)
Using Lagrange multipliers for imposing the boundary conditions leds to the
following equations
Π+ (U0 − Uref ) + Π− λ= 0,
Un+1 − Un Un+1 − Un+1
C 0 0
+A 1 0
+CΠ+ λ = 0;
∆t h
Un+1 − Un Un+1 − Un+1
C k k
+ A k+1 k−1
= 0, k ≥ 1.
∆t 2h
Π± is the projection operator onto incoming/outgoing waves, λ are the
Lagrange multipliers.
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25. Dynamic absorbing boundary conditions by M.Storti et.al.
Using penalization
Add a small regularization term and then eliminate the Lagrange multipliers.
− λ + Π+ (U0 − Uref ) + Π− λ = 0,
Un+1 − Un Un+1 − Un+1
C 0 0
+A 1 0
+ CΠ+ λ = 0;
∆t h
Eliminating the Lagrange multipliers λ we arrive to a boundary equation
Un+1 − Un Un+1 − Un+1
C 0 0
+A 1 0
+(1/ )CΠ+ (U0 − Uref ) = 0.
∆t h
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26. Dynamic absorbing boundary conditions by M.Storti et.al.
Absorbing boundary conditions and ALE
When using Arbitrary Lagrangian-Eulerian formulations:
∂U ∂Fc,j (U)
+ −vmesh U = 0
∂t ∂xj
∂Fc,j (U)
AALE,j = −vmesh,j I, ALE advective Jacobian
∂U
Nbr. of incoming characteristics = sum(eig(A · n) − vmesh · n < 0)
ˆ ˆ
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27. Dynamic absorbing boundary conditions by M.Storti et.al.
ALE invariance test case. Sudden stop of gas container
shock wave
expansion fan t t
x x
density
container initially moving container initially at rest
at constant speed u0 is suddenly put in movement
is suddenly stopped with constant negative speed −u0
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28. Dynamic absorbing boundary conditions by M.Storti et.al.
ALE invariance. SUPG Stabilization term
∂Uc ∂Fc,x ∂Fd,x
+ = ; (gov. eqs. in cons. form)
∂t ∂x ∂x
(13)
∂U ∂U ∂2U
C + A = K 2 ; (gov. eqs. in quasi-linear form)
∂t ∂x ∂x
˜
Sufficient conditions for ALE invariance (A = A − vmesh C)
P = ˜
N · Aτ C−1 ; (SUPG pert. function)
∂U ∂U (14)
τ transform as U × U, i.e. τ = τ .
∂U ∂U
˜
This last is verified if τ is f (C−1 A), for instance (inviscid case):
h ˜
τ = I, λj = eig(C−1 A), (max. eigenv.)
max |λj | (15)
˜
τ = h|C−1 A|−1 . (| · | in matrix sense)
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29. Dynamic absorbing boundary conditions by M.Storti et.al.
ALE invariance. Sudden stop of a gas container
γ = 1.4, u0 /c0 = 0.5. Results in both reference systems are equivalent to
machine precision.
1.2
pressure
1.0
x shock wave
0.8
0.6
1
0.4
0 t
0 A B C 1 D E 2 F G 3 4
expansion fan
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30. Dynamic absorbing boundary conditions by M.Storti et.al.
ALE invariance. Sudden stop of a gas container (cont.)
2
1.8
1.6 Eint Ek+E int
1.4
1.2
Energy
Ek
1
0.8
0.6
Ek
0.4
0.2
0
0 2 4 6 8 10 12 14
AB C D E F G t
2
(Up) Energy balance in
1.5
reference system fixed
w.r.t container)
Energy
1 Ek E W
(Down) Energy balance in int Ek+E int−W
reference system fixed
0.5
w.r.t initial gas at rest)
0
0 2 4 6 8 10 12 14
AB C D E F G
t
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31. Dynamic absorbing boundary conditions by M.Storti et.al.
Open channel flow.
Flow in a channel can be cast in advective form as follows
Up = [h, u]T ,
w
T
U = Uc = [A, Q] ,
H(U) = Uc ,
A h
Q
F = .
Q2 /A + F
where h and u are water depth and velocity (as in the shallow water
equations).
A(h) is the section of the channel occupied by water for a given water height
h. It then defines the geometry of the channel. Q = Au is the water flow rate.
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32. Dynamic absorbing boundary conditions by M.Storti et.al.
Open channel flow. (cont.)
For instance
Rectangular channels: A(h) = wh, w =width.
Triangular channels: A(h) = 2h2 tan θ/2; with
θ=angle opening.
Circular channel:
h
A(h) = 2Rh − h2 dh
h =0
2
= θR − w(h)(R − h)/2
√
w(h) = 2 2Rh − h2 ,
θ = atan[w/(2(R − h))] is angular aperture,
h
F (h) = h =0 A(h ) dh
Double rectangular channel.
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33. Dynamic absorbing boundary conditions by M.Storti et.al.
Comparison with RI for rectangular channel
|u_ulsar-u_RI| (at x=0)
|h_ulsar-h_RI| (at x=0)
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34. Dynamic absorbing boundary conditions by M.Storti et.al.
Comparison with RI for rectangular channel (cont.)
x=L (outgoing)
5.8
w+
5.7
5.6
5.5
right-going RI
5.4
5.3
5.2
5.1
0 2 4 6 8 10 12 14 16
x=0 (ingoing) time
-4.3
x=0 (outgoing)
w-
-4.4
-4.5
left-going RI
-4.6
x=L (ingoing)
-4.7
-4.8
0 2 4 6 8 10 12 14 16
time
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35. Dynamic absorbing boundary conditions by M.Storti et.al.
Comparison with RI for rectangular channel (cont.)
We restrict here to the case of
constant channel section and
depth.
For rectangular channels the
equations are reduced to those for
1D shallow water equations.
Channel flow is very interesting
since it is in fact a family of
different 1D hyperbolic systems
depending on the area function
A(h).
Riemann invariants are only known
for rectangular and triangular
channel shapes.
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36. Dynamic absorbing boundary conditions by M.Storti et.al.
Stratified shallow water flows
Another physical model where Riemann invariants have not a mathematical
closed form is the flow of a multi-layer fluid in channels. This kind of physical
model exists for instance when flow takes place on a mountainous terrain
over plain areas or dense distribution of torrents combined with heavy rainfall.
Each layer may have different density, velocity in a two-dimensional domain.
∂Uc ∂Fc,j (Uc ) ∂Uc
+ + Bj (Uc ) = G(Uc ); j = 1, 2. (16)
∂t ∂xj ∂xj
Uc = [h1 , h2 , h1 u1 , h1 v1 , h2 u2 , h2 v2 ]T is the vector of conservation
variables and Bj (Uc ) is the Jacobian matrix of the non-conservative
products in the j direction. h1 (x, t) and h2 (x, t) are the thickness of each
layer while the height of the bottom is h0 (x, t). [u, v]i is the velocity vector of
the layer i.
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38. Dynamic absorbing boundary conditions by M.Storti et.al.
Stratified shallow water flows
ρ1,2 are constant layer densities.
Vertical velocity is averaged.
Hydrostatic pressure assumption in each layer.
In the general case of n layers and 2D model, the system of equations has 3n
waves that propagate inside the domain.
It could happen that, depending on the densities ratio ρ2 /ρ1 , and the
velocities the system becomes non-hyperbolic. In fact if the velocities of the
fluid are the same, the flow is unstable for ρ2 > ρ1 .
Riemann-Invariants for stratified flow are not known.
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39. Dynamic absorbing boundary conditions by M.Storti et.al.
Stratified shallow water flows (cont.)
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40. Dynamic absorbing boundary conditions by M.Storti et.al.
Acknowledgment
This work has received financial support from Consejo Nacional de
´
Investigaciones Cient´ficas y Tecnicas (CONICET, Argentina, PIP 5271/05),
ı
Universidad Nacional del Litoral (UNL, Argentina, grants CAI+D 2005-10-64)
´ ´
and Agencia Nacional de Promocion Cient´fica y Tecnologica (ANPCyT,
ı
Argentina, grants PICT PME 209/2003, PICT-1141/2007, PICT-1506/2006).
We made extensive use of Free Software (http://www.gnu.org) as
GNU/Linux OS, MPI, PETSc, GCC/G++ compilers, Octave, Open-DX, VTK,
Python, Git, among many others. In addition, many ideas from these packages
have been inspiring to us.
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