Finite Difference Method Numerical solution of Laplace Equation using MATLAB. 2 computational methods are used.
U can vary the number of grid points and the boundary conditions
Numerical Solution for Two Dimensional Laplace Equation with Dirichlet Bounda...IOSR Journals
In this paper numerical technique has been used to solve two dimensional steady heat flow problem with Dirichlet boundary conditions in a rectangular domain and focuses on certain numerical methods for solving PDEs; in particular, the Finite difference method (FDM), the Finite element method (FEM) and Markov chain method (MCM) are presented by using spreadsheets. Finally the numerical solutions obtained by FDM, FEM and MCM are compared with exact solution to check the accuracy of the developed scheme
Mathematics and History of Complex VariablesSolo Hermelin
Mathematics of complex variables, plus history.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com, thanks! For more presentations, please visit my website at http://www.solohermelin.com
How to handle Initial Value Problems using numerical techniques?
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Initial+Value+Problems
https://eau-esa.wikispaces.com/Topic+Initial+Value+Problems
Numerical Solution for Two Dimensional Laplace Equation with Dirichlet Bounda...IOSR Journals
In this paper numerical technique has been used to solve two dimensional steady heat flow problem with Dirichlet boundary conditions in a rectangular domain and focuses on certain numerical methods for solving PDEs; in particular, the Finite difference method (FDM), the Finite element method (FEM) and Markov chain method (MCM) are presented by using spreadsheets. Finally the numerical solutions obtained by FDM, FEM and MCM are compared with exact solution to check the accuracy of the developed scheme
Mathematics and History of Complex VariablesSolo Hermelin
Mathematics of complex variables, plus history.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com, thanks! For more presentations, please visit my website at http://www.solohermelin.com
How to handle Initial Value Problems using numerical techniques?
#WikiCourses
https://wikicourses.wikispaces.com/Topic+Initial+Value+Problems
https://eau-esa.wikispaces.com/Topic+Initial+Value+Problems
ME-314 Introduction to Control Engineering is a course taught to Mechanical Engineering senior undergrads. The course is taught by Dr. Bilal Siddiqui at DHA Suffa University. This lecture is about modeling electrical and mechanical systems (transnational and rotational) in frequency domain.
Subject Title: Engineering Numerical Analysis
Subject Code: ID-302
Contents of this chapter:
Mathematical preliminaries,
Solution of equations in one variable,
Interpolation and polynomial Approximation,
Numerical differentiation and integration,
Initial value problems for ordinary differential equations,
Direct methods for solving linear systems,
Iterative techniques in Matrix algebra,
Solution of non-linear equations.
Approximation theory;
Eigen values and vector;
Multi Objective Optimization and Pareto Multi Objective Optimization with cas...Aditya Deshpande
This gives basic idea of MOO ie. Multi Objective Optimization and also Pareto graph used for it.
Here i have done Ansys optimization on simple object to elaborate concept of MOO.
Thanks
Aditya D
deshadi805@gmail.com
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.
Jacobi Iteration Method is Used in Numerical Analysis. This slide helps you to figure out the use of the Jacobi Iteration Method to submit your presentatio9n slide for academic use.
ME-314 Introduction to Control Engineering is a course taught to Mechanical Engineering senior undergrads. The course is taught by Dr. Bilal Siddiqui at DHA Suffa University. This lecture is about modeling electrical and mechanical systems (transnational and rotational) in frequency domain.
Subject Title: Engineering Numerical Analysis
Subject Code: ID-302
Contents of this chapter:
Mathematical preliminaries,
Solution of equations in one variable,
Interpolation and polynomial Approximation,
Numerical differentiation and integration,
Initial value problems for ordinary differential equations,
Direct methods for solving linear systems,
Iterative techniques in Matrix algebra,
Solution of non-linear equations.
Approximation theory;
Eigen values and vector;
Multi Objective Optimization and Pareto Multi Objective Optimization with cas...Aditya Deshpande
This gives basic idea of MOO ie. Multi Objective Optimization and also Pareto graph used for it.
Here i have done Ansys optimization on simple object to elaborate concept of MOO.
Thanks
Aditya D
deshadi805@gmail.com
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.
Jacobi Iteration Method is Used in Numerical Analysis. This slide helps you to figure out the use of the Jacobi Iteration Method to submit your presentatio9n slide for academic use.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
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Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
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Governing Equations for Fundamental Aerodynamics_Anderson2010.pdf
FDM Numerical solution of Laplace Equation using MATLAB
1. Numerical Analysis
Visit my BlogSpot
http://ayaozaki.blogspot.com/2014/06/
fdm-numerical-solution-of-laplace.html
6/25/2014 Aya Zaki 1
2. A Finite Difference Method for
Laplace’s Equation
• A MATLAB code is introduced to solve Laplace
Equation.
• 2 computational methods are used:
– Matrix method
– Iteration method
• Advantages of the proposed MATLAB code:
– The number of the grid point can be freely chosen
according to the required accuracy.
– The boundary conditions can simply be changed.
6/25/2014 Aya Zaki 2
3. A Finite Difference Method for
Laplace’s Equation (cont.)
• Example (Sheet 4)
• Grid: N=3
• B.C. shown
6/25/2014 Aya Zaki 3
50
37.5
25
12.5
37.52512.5
000
0
0
0
0
0
200mm
200mm
T(i,j)
5. U = inv(A)*B;
%Re-arrange
for j= 1:N
for i=1:N
T(j+1,i+1)= U((j-
1)*N+i);
end
end
for i= 1:N
for j=1:N
k= (j-1)*N +i;
A(k,k)= -4;
for m = i-1: 2:i+1
if ((m<1) ||(m>N))
B(k)= B(k) -T(m+1,j+1);
else
l = (j-1)*N+m;
A(k,l)= 1;
end
end
for n = j-1: 2:j+1
if ((n<1) ||(n>N))
B(k)= B(k)- T(i+1,n+1);
else
l = (n-1)*N+i;
A(k,l)= 1;
end
end
end
end
I. Matrix computation method(cont.)
• Code
6/25/2014 Aya Zaki 5
N=3;
T = zeros(N+2, N+2);
x = linspace(0,200e-3, N+2);
y = linspace(0,200e-3, N+2);
%Boundary Conditions
% Y- left
T(:,N+2) = linspace(0,50,N+2)
% Top
T(N+2,:)= linspace(0,50,N+2)
A , B
9. • N= 50
I. Matrix computation method(cont.)
6/25/2014 Aya Zaki 9
0
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0.2
0
10
20
30
40
50
Distance x
Numerical solution computed with solving the Matrix.
Distance y
10. for k=1:20
for i=2:N+1
for j=2:N+1
un(i,j)=(un(i+1,j)+un(i-1,j)+un(i,j+1)+un(i,j-1))/4;
end
end
end
II. Iteration computation method
• Code
– Number of iterations = 20
– The better the initial guess, the faster the computation is.
– For simplicity, the initial value for all points is chosen as zero.
6/25/2014 Aya Zaki 10
N=3;
T = zeros(N+2, N+2);
x = linspace(0,200e-3, N+2);
y = linspace(0,200e-3, N+2);
%Boundary Conditions
% Y- left
T(:,N+2) = linspace(0,50,N+2)
% Top
T(N+2,:)= linspace(0,50,N+2)
12. • Plotting T
II. Iteration computation method
(cont.)
6/25/2014 Aya Zaki 12
0
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0.2
0
10
20
30
40
50
Distance x
Numerical solution computed with 20 iteration.
Distance y
The same results were obtained as before.
0
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0.2
0
10
20
30
40
50
13. • Plotting T
II. Iteration computation method
(cont.)
6/25/2014 Aya Zaki 13
0
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0.2
0
10
20
30
40
50
Distance x
Numerical solution computed with 20 iteration.
Distance y
The same results were obtained as before.
0
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0.2
0
10
20
30
40
50
0
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0.2
0
10
20
30
40
50
14. • Plotting T
II. Iteration computation method
(cont.)
6/25/2014 Aya Zaki 14
0
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0.2
0
10
20
30
40
50
Distance x
Numerical solution computed with 20 iteration.
Distance y
The same results were obtained as before.
15. A Finite Difference Method for
Laplace’s Equation (cont.)
• Example (Sheet 4)
• Grid: N=3
• B.C. with lower
edge insulated.
6/25/2014 Aya Zaki 15
50
37.5
25
12.5
37.52512.5
000
0
0
0
0
0
200mm
200mm
T(i,j)