On Approach to Increase Integration Rate of Elements of a Switched-capacitor ...BRNSS Publication Hub
In this paper, we introduce an approach to increase integration rate of elements of a switched-
capacitor step-down DC–DC converter. Framework the approach, we consider a heterostructure with
special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion
implantation. Annealing of dopant and/or radiation defects should be optimized.
On Approach to Increase Integration rate of Elements of a Circuit Driver with...BRNSS Publication Hub
In this paper, we introduce an approach to increase the integration rate of elements of a driver with 2-tap de-emphasis and impendence matching. Framework the approach, we consider a heterostructure with special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radiation defects should be optimized.
On Approach to Increase Integration Rate of Elements of a Current Source CircuitBRNSS Publication Hub
In this paper, we introduce an approach to increase integration rate of elements of a current source circuit.
Framework the approach, we consider a heterostructure with special configuration. Several specific
areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/
or radiation defects should be optimized.
An Approach to Analyze Non-linear Dynamics of Mass Transport during Manufactu...BRNSS Publication Hub
In this paper, we introduce an approach to increase integration rate of elements of a hybrid comparator with the first dynamic amplifying stage and the second quasi-dynamic latching stage. Framework the approach, we consider a heterostructure with special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radiation defects should be optimized
INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED-HETEROJUNCTION RECTIFIERSZac Darcy
In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a
heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the
considered p-n-junction have been formulated.
On Approach to Increase Integration Rate of Elements of a Switched-capacitor ...BRNSS Publication Hub
In this paper, we introduce an approach to increase integration rate of elements of a switched-
capacitor step-down DC–DC converter. Framework the approach, we consider a heterostructure with
special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion
implantation. Annealing of dopant and/or radiation defects should be optimized.
On Approach to Increase Integration rate of Elements of a Circuit Driver with...BRNSS Publication Hub
In this paper, we introduce an approach to increase the integration rate of elements of a driver with 2-tap de-emphasis and impendence matching. Framework the approach, we consider a heterostructure with special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radiation defects should be optimized.
On Approach to Increase Integration Rate of Elements of a Current Source CircuitBRNSS Publication Hub
In this paper, we introduce an approach to increase integration rate of elements of a current source circuit.
Framework the approach, we consider a heterostructure with special configuration. Several specific
areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/
or radiation defects should be optimized.
An Approach to Analyze Non-linear Dynamics of Mass Transport during Manufactu...BRNSS Publication Hub
In this paper, we introduce an approach to increase integration rate of elements of a hybrid comparator with the first dynamic amplifying stage and the second quasi-dynamic latching stage. Framework the approach, we consider a heterostructure with special configuration. Several specific areas of the heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radiation defects should be optimized
INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED-HETEROJUNCTION RECTIFIERSZac Darcy
In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a
heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the
considered p-n-junction have been formulated.
Identification of the Mathematical Models of Complex Relaxation Processes in ...Vladimir Bakhrushin
The approach to solving the problem of complex relaxation spectra is presented.
Presentation for the XI International Conference on Defect interaction and anelastic phenomena in solids. Tula, 2007.
On Approach to Increase Integration Rate of Elements of an Operational Amplif...BRNSS Publication Hub
In this paper, we introduce an approach to optimize manufacturing of an operational amplifier circuit based on field-effect transistors. Main aims of the optimization are (i) decreasing dimensions of elements of the considered operational amplifier and (ii) increasing of performance and reliability of the considered field-effect transistors. Dimensions of considered field-effect transistors will be decreased due to manufacture of these transistors framework heterostructure with specific structure, doping of required areas of the heterostructure by diffusion or ion implantation, and optimization of annealing of dopant and/or radiation defects. Performance and reliability of the above field-effect transistors could be increased by optimization of annealing of dopant and/or radiation defects and using inhomogeneity of properties of heterostructure. Choosing of inhomogeneity of properties of heterostructure leads to increasing of compactness of distribution of concentration of dopant. At the same time, one can obtain increasing of homogeneity of the above concentration. In this paper, we also introduce an analytical approach for prognosis of technological process of manufacturing of the considered operational amplifier. The approach gives a possibility to take into account variation of parameters of processes in space and at the same time in space. At the same time, one can take into account nonlinearity of the considered processes.
Lyapunov-type inequalities for a fractional q, -difference equation involvin...IJMREMJournal
In this paper, we present new Lyapunov-type inequalities for a fractional boundary value problem of
fractional
q, -difference equation with p-Laplacian operator. The obtained inequalities are used to obtain a
lower bound for the eigenvalues of corresponding equations.
Positive and negative solutions of a boundary value problem for a fractional ...journal ijrtem
: In this work, we study a boundary value problem for a fractional
q, -difference equation. By
using the monotone iterative technique and lower-upper solution method, we get the existence of positive or
negative solutions under the nonlinear term is local continuity and local monotonicity. The results show that we
can construct two iterative sequences for approximating the solutions
An Approach to Optimize Regimes of Manufacturing of Complementary Horizontal ...ijrap
In this paper we consider nonlinear model to describe manufacturing complementary horizontal field-effect heterotransistor. Based on analytical solution of the considered boundary problems some recommendations have been formulated to optimize technological processes.
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
We formulate the initial value problem to model the evolution of the interface between two fluids of different density in three spatial dimensions. The evolution equations account for the action of gravity on the fluids, surface tension in the fluids and a prescribed far-field conditions.
The flow in each fluid is incompressible and irrotational, so the classical potential theory applies and allows for a boundary integral of dipoles representation. This representation satisfies the kinematic condition of continuous normal velocity and the Laplace-Young condition for the pressure. The dipole strength is related to the jump in potential across the interface. The model of the exact nonlinear three-dimensional motion of the interface is formulated and includes expressions for integral invariants of the motion, the mean height of the interface and the total energy per wavelength.
We develop the numerical method that employes a special generalized isothermal interface parameterization. It enables the use of implicit non-stiff time-integration methods via a small-scale decomposition. Our method includes the efficient algorithms for the generation of initial data with the generalized isothermal parameterization by evolving a flat interface toward a prescribed initial surface shape or by the appropriate choice of the tangential velocities.
The method is used to efficiently compute the nonlinear evolution of a doubly periodic interface separating two fluids in the Rayleigh-Taylor instability and internal waves with surface tension.
Identification of the Mathematical Models of Complex Relaxation Processes in ...Vladimir Bakhrushin
The approach to solving the problem of complex relaxation spectra is presented.
Presentation for the XI International Conference on Defect interaction and anelastic phenomena in solids. Tula, 2007.
On Approach to Increase Integration Rate of Elements of an Operational Amplif...BRNSS Publication Hub
In this paper, we introduce an approach to optimize manufacturing of an operational amplifier circuit based on field-effect transistors. Main aims of the optimization are (i) decreasing dimensions of elements of the considered operational amplifier and (ii) increasing of performance and reliability of the considered field-effect transistors. Dimensions of considered field-effect transistors will be decreased due to manufacture of these transistors framework heterostructure with specific structure, doping of required areas of the heterostructure by diffusion or ion implantation, and optimization of annealing of dopant and/or radiation defects. Performance and reliability of the above field-effect transistors could be increased by optimization of annealing of dopant and/or radiation defects and using inhomogeneity of properties of heterostructure. Choosing of inhomogeneity of properties of heterostructure leads to increasing of compactness of distribution of concentration of dopant. At the same time, one can obtain increasing of homogeneity of the above concentration. In this paper, we also introduce an analytical approach for prognosis of technological process of manufacturing of the considered operational amplifier. The approach gives a possibility to take into account variation of parameters of processes in space and at the same time in space. At the same time, one can take into account nonlinearity of the considered processes.
Lyapunov-type inequalities for a fractional q, -difference equation involvin...IJMREMJournal
In this paper, we present new Lyapunov-type inequalities for a fractional boundary value problem of
fractional
q, -difference equation with p-Laplacian operator. The obtained inequalities are used to obtain a
lower bound for the eigenvalues of corresponding equations.
Positive and negative solutions of a boundary value problem for a fractional ...journal ijrtem
: In this work, we study a boundary value problem for a fractional
q, -difference equation. By
using the monotone iterative technique and lower-upper solution method, we get the existence of positive or
negative solutions under the nonlinear term is local continuity and local monotonicity. The results show that we
can construct two iterative sequences for approximating the solutions
An Approach to Optimize Regimes of Manufacturing of Complementary Horizontal ...ijrap
In this paper we consider nonlinear model to describe manufacturing complementary horizontal field-effect heterotransistor. Based on analytical solution of the considered boundary problems some recommendations have been formulated to optimize technological processes.
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
We formulate the initial value problem to model the evolution of the interface between two fluids of different density in three spatial dimensions. The evolution equations account for the action of gravity on the fluids, surface tension in the fluids and a prescribed far-field conditions.
The flow in each fluid is incompressible and irrotational, so the classical potential theory applies and allows for a boundary integral of dipoles representation. This representation satisfies the kinematic condition of continuous normal velocity and the Laplace-Young condition for the pressure. The dipole strength is related to the jump in potential across the interface. The model of the exact nonlinear three-dimensional motion of the interface is formulated and includes expressions for integral invariants of the motion, the mean height of the interface and the total energy per wavelength.
We develop the numerical method that employes a special generalized isothermal interface parameterization. It enables the use of implicit non-stiff time-integration methods via a small-scale decomposition. Our method includes the efficient algorithms for the generation of initial data with the generalized isothermal parameterization by evolving a flat interface toward a prescribed initial surface shape or by the appropriate choice of the tangential velocities.
The method is used to efficiently compute the nonlinear evolution of a doubly periodic interface separating two fluids in the Rayleigh-Taylor instability and internal waves with surface tension.
My presentation at University of Nottingham "Fast low-rank methods for solvin...Alexander Litvinenko
Overview of my (with co-authors) low-rank tensor methods for solving PDEs with uncertain coefficients. Connection with Bayesian Update. Solving a coupled system: stochastic forward and stochastic inverse.
We have implemented a multiple precision ODE solver based on high-order fully implicit Runge-Kutta(IRK) methods. This ODE solver uses any order Gauss type formulas, and can be accelerated by using (1) MPFR as multiple precision floating-point arithmetic library, (2) real tridiagonalization supported in SPARK3, of linear equations to be solved in simplified Newton method as inner iteration, (3) mixed precision iterative refinement method\cite{mixed_prec_iterative_ref}, (4) parallelization with OpenMP, and (5) embedded formulas for IRK methods. In this talk, we describe the reason why we adopt such accelerations, and show the efficiency of the ODE solver through numerical experiments such as Kuramoto-Sivashinsky equation.
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.
Representation of of Stochastic Processes in Stochastic Processes in Real and Spectral Domains Real and Spectral Domains and and Monte Monte-Carlo sampling
Nondeterministic testing of Sequential Quantum Logic Propositions on a Quant...Matthew Leifer
Talk given at "Quantum Information, Computation and Logic" workshop at Perimeter Institute in 2005.
Based on the preprint http://arxiv.org/abs/quant-ph/0509193
Talk was recorded and is viewable online at http://pirsa.org/05070102/
I now regard these ideas as flawed, but they may be revisited in future work.
Aerodynamic and Acoustic Parameters of a Coandã Flow – a Numerical Investigationdrboon
Coandã flows have been the study of aircraft designers primarily for the prospect of achieving higher lift coefficient wings. Recently the environmental problem of noise pollution attracted further interest on the matter. The approach used is numerical; the computations were made using a large eddy simulation (LES) technique coupled with a Ffowcs-Williams-Hawkings (FWH) acoustic analysis. The spectrum of the flow was measured at three locations in the vicinity of the ramp showing that the low frequency region is dominant. The findings may be used as reference for the development of quiet aircraft that use super-circulation, as it is the case with the Upper Surface Blown (USB) configurations.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
Let's dive deeper into the world of ODC! Ricardo Alves (OutSystems) will join us to tell all about the new Data Fabric. After that, Sezen de Bruijn (OutSystems) will get into the details on how to best design a sturdy architecture within ODC.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
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/
1. Department of Mechanical Engineering, The University of British Columbia
A Higher Order Accurate Unstructured Finite Volume
Higher-Order Finite-Volume
Newton-Krylov Algorithm for Inviscid Compressible Flows
Amir Nejat
Knowledge Diffusion Network
١٣٨۶ داﻧﺸﮑﺪﻩ ﻣﻬﻨﺪﺳﯽ هﻮاﻓﻀﺎ، داﻧﺸﮕﺎﻩ ﺻﻨﻌﺘﯽ ﺷﺮﻳﻒ، ٩٢ﻣﻬﺮﻣﺎﻩ
2. Aircraft Design & Fuel Efficiency
η : Fuel consumption per seat per mile
η 777 < η 767 15%
η 787 < η 777 20%
3. Design Process
Mission Specification
Experience Initial Design
Multi-Physics Numerical Multi-Disciplinary
PDE S l
Solvers Optimization
Optimized Design
Opening: Design Process CFD
4. CFD
1-Mesh
Complex Geometry
Adaptation and Refinement
2-Accuracy
Discretization (Truncation) error
Modeling error
3-Convergence
3C
Stability
Residual dropping order
Time & Cost
Background: CFD CFD Algorithm
5. CFD - Overall Algorithm
Geometry & Solution domain Mesh generation package
Physics & Fluid flow equations
Meshed domain
Residual
Boundary & Initial conditions
Discretization of the fluid flow equations
& Flux Computation and Integration
Implicit method
L
Large system of li
t f linear equations
ti
Jacobian matrix
Sparse Fluid flow
Preconditioning
matrix solver
simulation
Background: CFD Algorithm Motivation
6. Motivation
∂U ∂U
Second-order methods: U 2 nd −order= U ( xc , yc ) + Δx + Δy + O( Δ )2
∂x ∂y
∂ 2U Δx 2 ∂ 2U ∂ 2U Δy 2
Truncation error: O( Δ ) = 22
+ ΔxΔy + 2
∂x 2 ∂x∂y ∂y 2
The 2nd-order truncation error acts like a diffusive term and causes
two significant numerical problems:
1-It smears sharp gradients and spoils total pressure conservation (isentropic flows).
2-It produces parasitic error by adding extra diffusion to viscous regions.
Higher-order: More accurate simulation
Existing research shows higher-order structured discretization technique for a
given level of accuracy is more efficient.
Higher-order:
Higher order: Can be more efficient !?
Background: Motivation Literature Review
7. Literature Review
Qualitative Illustration of Research on Solver Development
Structured Structured-Implicit Unstructured Unstructured-Implicit
Second-order
♣♣♣♣♣♣♣♣♣ ♣♣♣♣ ♣♣♣♣♣♣ ♣♣♣
Higher-order
♣♣♣ ♣♣ ♣ ?
Trend:
1- Increasing the efficiency using convergence acceleration techniques
such as implicit methods (Newton-Krylov).
2- Enhancing the accuracy using higher-order discretization scheme.
Background: Literature Review Contribution
8. Objective
• Developing an Efficient Higher-Order Accurate
Unstructured Finite Volume Algorithm for Inviscid
Compressible Fluid Flow.
Objective: Contribution Model Problem
9. Model Problem
The unsteady (2D) Euler equations which model compressible inviscid
fluid flows, are conservation equations for mass, momentum, and energy.
Aerodynamic application: lift, wave drag and induced drag
d
∫ Udv + ∫ FdA = 0
dt cv
(1)
cs
⎡ρ⎤ ⎡ ρun ⎤
⎢ ρu ⎥ ⎢ ρuu + Pn ⎥ˆx
U =⎢ ⎥ , F =⎢ n ⎥ (2)
⎢ ρv ⎥ ⎢ ρvun + Pn y ⎥
ˆ
⎢ ⎥ ⎢ ⎥
⎣E⎦ ⎣ ( E + P )un ⎦
u n = un x + vn y , E = P /( γ − 1 ) + ρ (u 2 + v 2 ) / 2
ˆ ˆ
Theory: Model Problem Implicit Time Advance
10. Implicit Time Advance
Applying implicit time integration and linearization of the governing
equations in time leads to implicit time advance formula:
dU U n +1 − U n
( + R( U ) ) = 0 ⇒ ( + R n +1 ) = 0 (3)
dt Δt
n +1 ∂R n n+1
R = Rn + ( ) (U −U n ) (4)
∂U
I ∂R
( + )δU = − R , δU = U n+1 − U n
n
(5)
Δt ∂U
U: Solution Vector
R: Residual Vector
∂R/∂U: Jacobian matrix
Eq. 5 is a system of linear equations arising from discretization of
governing equations over unstructured domain.
Theory: Implicit Time Advance Linear System Solver
11. Linear System Solver
GMRES (Generalized Minimal Residual, Saad 1986)
*GMRES algorithm, among other Krylov techniques, only needs matrix vector
d t ( t i f
products (matrix-free i limplementation).
t ti )
*It is developed for non-symmetric matrices.
*It predicts the best solution update if the linearization is carried out accurately.
To enhance the convergence performance of the GMRES solver, it is necessary to
apply preconditioning:
−1
Ax = b − > ( AM ) Mx = b , A ≈ M
M = LU
M ≅ ILU ( n )
M is an approximation to matrix A which has simpler structure.
ILU: Incomplete Lower-Upper factorization
p pp
Technique: Linear System Solver Reconstruction
12. Reconstruction
• Defining the Kth-order polynomial for each control
volume.
• Finding the polynomial coefficients using the averages of
the neighboring control volumes.
• This polynomial is constructed based on some constraints
such as mean constraint.
h t i t
∂U ∂U
= U ( xc , yc ) + Δx + Δy +
(K)
UR
∂x ∂y
∂ 2U Δx 2 ∂ 2U ∂ 2U Δy 2
+ ΔxΔy + 2 +
∂x 2 2 ∂x∂y ∂y 2
∂ 3U Δx 3 ∂ 3U Δx 2 Δy ∂ 3U ΔxΔy 2 ∂ 3U Δy 3
+ 2 + + 3 + ... (6) ∫U R
(K)
( x , y ) = U CV (7)
∂x 63
∂x ∂y 2 ∂x∂y 2
2 ∂y 6 CV
Technique: Reconstruction Monotonicity
13. Monotonicity
Limiting
Limiting
g
Technique: Monotonicity Higher-Order Limiter
15. Flux Evaluation
• Discretization scheme :
Solution reconstruction: Kth-order accurate least-square
reconstruction procedure (Ollivier-Gooch 1997)
t ti d (Olli i G h 1997).
Flux formulation: Roe’s flux difference splitting (1981).
1 1 ~
F (U L ,U R ) = ( F (U L ) + F (U R )) − A (U R − U L ) (10)
2 2 ( L, R )
~ ~ ~ ~ ~ ~
A = X −1 Λ X , Λ = Diag λ
• Integration scheme : Gauss quadrature integration technique
with the proper number of p
p p points.
Ri = ∫ F .nds
CVi
(11)
Gauss quadrature for interior control volumes.
Technique: Flux Evaluation 1st-Order Jacobian Matrix
16. 1st-Order Jacobian Matrix
Ri = ∑ F nds = ∑ F ( U ,U
ˆ
faces
i i Nk
ˆ
)( nl )i ,N k (12)
∂Ri ∂F ( U i ,U N k )
J ( i, Nk ) = = ˆ
( nl )i ,N k (13-1)
∂U N k ∂U N k
∂Ri ∂F ( U i ,U N k )
J ( i ,i ) = =∑ ˆ
( nl )i ,N k (13-2)
∂U i ∂U i
Technique: 1st-Order Jacobian Matrix Solution Strategy
18. Solution Procedure
• Start up Process :
Before switching to Newton-GMERS Iteration, several pre-implicit
iterations have been performed in the form of defect correction, using
Eq. (5).
I ∂R
( + )δU = − R (5)
Δt ∂U
∂R
(First Order)
∂U
Resultant system is solved by GMRES - ILU(1) linear solver.
• Newton-GMRES (matrix-free) iteration :
At this stage, infinite time step is taken, and GMRES-ILU(4) is used to
g , p , ( )
solve the linear system at each Newton iteration.
∂R ∂R R( U + εv ) − R( U )
( )δU = − R (12) .v ≅ (13)
∂U ∂U ε
Procedure: Solution Procedure Results
19. Results
Supersonic Vortex, Annulus-Meshes
p ,
427 CVs 1703 CVs
108 CV
CVs
6811 CVs 27389 CVs
Results: Supersonic Vortex Mach Contours Density Error Error Convergence Error versus CPU Time
23. Density Error versus CPU Time / Supersonic Vortex,
M 2.0
M=2.0
Results: Error versus CPU Time Subsonic flow over NACA 0012 Airfoil Subsonic Convergence
24. Subsonic Flow over NACA 0012, M=0.63, AoA=2.0 deg.
4958CV 2nd-Order
3rd-Order
Order 4th-Order
Order
25. Convergence history-Subsonic Case
Order Resid. Eval. Time (Sec) Work Units Newton Itr. Newton Work Units
2nd 126 26.88 349.1 3 136.1-39%
3rd 147 36.03 248.5 4 141.2-57%
4th 247 90.54
90 54 289.3
289 3 7 239.2-83%
239 2-83%
Results: Subsonic Convergence Transonic flow over NACA 0012 Airfoil Transonic Convergence
27. Convergence history-Transonic Case
Order Resid. Eval. Time (Sec) Work Units Newton Itr. Newton Work Units
2nd 197 65.6 279 4 91-33%
3rd 241 106.7 281 5 119-42%
4th 450 311 4
311.4 590 10 221-37%
Results: Transonic Convergence Transonic Mach Profile
28. Mach Profile-Transonic case
Order CL CD
2nd 0.337593 0.0220572
3rd 0.339392 0.0222634
4th 0.345111 0.0224720
AGARD / Structured (7488:192*39) 0.3474 0.0221
Results: Transonic Mach Profile Research Summary and Conclusion
29. Research Summary and Conclusion
• An ILU preconditioned GMRES algorithm (matrix-free) has been used for
efficient higher-order computation of solution of Euler equations.
• A start-up procedure is implemented using defect correction pre-iterations
before switching to Newton iterations.
• As an over all performance assessment (including the start up phase) the third
start-up
order solution is about 1.3 to 1.5 times, and the fourth order solution is about
3.5-5 times, more expensive than the second order solution with the developed
gy
solver technology.
• A modified Venkatakrishnan Limiter was implemented to address the
convergence hampering issue, and to improve the accuracy of the limited
eco s uc o .
reconstruction.
• Using a good initial solution state, start up process and effective
preconditioning are determining factors in Newton-GMRES solver
performance
performance.
• The possibility of benefits of higher-order discretization has been shown.
Closing: Research Summary and Conclusion Recommended Future Work
30. Recommended Future Work
• Improving the start-up procedure.
• Applying a more accurate preconditioning.
pp y g p g
• Enhancing th b t
E h i the robustness of the reconstruction f di
f th t ti for discontinuities (limiting).
ti iti (li iti )
• Extension to 3D.
• Extension to viscous flows.
Closing: Recommended Future Work End