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I want to learn the basics of B-spline curve in order to better understand the NURBS.
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Thanks for raising such an interesting question. Probably some people would be interested in the following special issue.
"Modern Geometric Modeling: Theory and Applications II" (IF=2.258) (Deadline: February 28, 2022)
The scope of the Special Issue includes but is not limited to original research works within the subject of geometric modeling and its applications in engineering, arts, physics, biology, medicine, computer graphics, architecture, etc., as well as theoretical mathematics and geometry which can be applied to problems of geometric modeling. For this Special Issue, we plan to accept the following types of manuscripts:
  • Overviews;
  • Research manuscripts;
  • Short manuscripts which discuss open problems in geometric modeling.
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Hello,
Suppose we know the quaternions describing orientation of PIP,DIP and MCP respectively at every instant of time, how can we use the same to estimate the joint angle ie, how much flexion/extension, abduction/adduction ? 
I know relative orientation can be found and Euler angles can be decomposed if we know the euler angle sequence or finding the rotation matrices for respective segement and using the dot product of the aligned axes (not sure this part).
Can anyone suggest a method of finding the joint angle or reading material to find the same.
Thanks and Regards,
Akhil
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Hi Mathias,
Please find the i-Glove paper "An instrumented glove for monitoring hand function: Review of Scientific Instruments: Vol 89, No 10." for more information.
Regards,
Akhil
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Hello, guys,
I performed a simulation with modeled system, and I put the input signal as an sine wave, with frequency varies from [1:1:1500] Hz.
(In detail, I performed simulation with sine input of fixed frequency for 2 seconds, and repeat this for every 1500 cases.)
As a result of simulation, I got 1500 pairs of input & output data.
I know how to get T.F. from impulse response but I don't know how to do it with sinusoidal waves.. 
Can anyone explain the process to get T.F for my case? 
Thanks.
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Hi,
Now you have the input and output data in sine wave form.
Obtain the FFT of input data and the output data.
Divide the FFT (Output) with the FFT(input)
let's say it's now FFT(Op)/FFT(ip) = x. Then by making use of the Real and Imag function find Real(x) and Imag(x). Then plot them as x axis as real and y as imaginary.
Now, you have the nyquist plot of the system from where you can easily find the poles and zeros and finally your transfer function.
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I Have Digital maps in PDF files and I would like to convert them into either Vector formats or Raster Formats. Any idea or any software that can help me?
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Yet another alternative converter: https://pdftosvg.com/
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Hi all,
I have a basic doubt in linear algebra. The determinant of a 2x2 matrix can be considered as the area of a parallelogram. Similarly what could be the physical interpretation of a characteristic equation  and roots of the characteristic Equation (Eigen Value)?
This question is bugging my mind for a long time now. Can any one here enlighten me?
Regards,
Balaji
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Mathematically, eigenvalues can interpret the stability of your system depending on the sign of your eigenvalues. 
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Hi
in ansys fluent's dacuments it has been said that : (( ANSYS Fluent has taken steps to offer more advanced wall formulations, which
allow a consistent mesh refinement without a deterioration of the results. Such -independent formulations are the default for
all omega-equation-based turbulence models. For the epsilon-equation-based models, the Menter-Lechner and Enhanced Wall Treatment (EWT)
serve the same purpose. A -insensitive wall treatment is also the default for the Spalart-Allmaras model and allows you to run this model
independent of the near-wall resolution. ))
so, is it true that for models mentioned above like k-omega or k-epsilon(EWT,Menter-lechner)there is no need to check y+ value to be in a certain range ?
for example can i use k-epsilon model with enhanced wall treatment and y+ value changes between 0.5 to 600 ?
if my assumption is'nt true , what is the certain range of y+ value for all turbulence models like k-epsilon with wall functions or k-omega
without wall functions ? please give me a reliable source ...
thanks...
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Dear Amir,
Let me give you some more explanation concerning y plus. Y plus is an analytical profile covering the viscous sublayer and the buffer layer. In fact, the boundary layer thickness composed of three regions when going far from the wall, started with viscous sublayer, the buffer layer, and the turbulent layer. Y plus is designed to cover only the first two parts and it is expected to be started from the very beginning of the turbulent layer. Now as an assumption keep these figures in your mind:
Yplus> 500 for the outer layer
31<Yplus<500 for log low region
5<Yplus<30 for the buffer layer
Yplus<5 for viscous sublayer
considering above applying the wrong value for Yplus could shift the calculation from one layer to the other, where it is not supposed to be. In this regard, you need to run your model first by a fix y plus let say 50 for all parts of the model, thereupon check the model in postprocessing and determine the regions with high-velocity gradient, calculate boundary layer again for local Reynolds value, then you would realize how much YPlus value you need for different region and where you need to reduce it and up to what figure. In other words, reducing y plus means that regions are exposed to much more intensified turbulent flow, and thereof you need to close the grid to the wall. If we suppose sigma as the boundary layer thickness the ration of 0.02<y/sigma<0.2 represents the region one needs to locate the starting grid.
To put it in another way, the best solution could be obtained without using Y plus and by direct meshing right from the wall surface, but we made use of y plus to decrease computational costs associated with direct meshing.
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Many times after the paper is published, the authors note that a major flaw has taken place in published work. . The flaw can either be in formulating the problem or in the methodology or assumptions or errors in measurement. The published results are totally wrong. In such cases, do authors on their own accept the mistake and withdraw the published paper?
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The researcher should notify the editor-in-chief, and the author should fix it if possible.
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Dear All,
I modeled a rotating blade in the Abaqus/CAE 6.14-2 x64. The blade is generated using the properties of different sections of the airfoils along the span. The generalized beam section is used for different sections of the blade. The mass of the blade is given as a longitudinal mass in line with its length. The material used for blade has an elastic modulus of 13.1 GPa, and a shear modulus of 8 GPa. The properties of this generalized section consisted of the area (A), the moment of inertia along two axes (I11, I22), and polar moment of inertia (J). Element type B31 was used to meshing the blade.
The blade was then spun by the Displacement/Rotation boundary condition of Abaqus around an axis through one end perpendicular to its length. The rotation was modeled in both Static-General and Dynamic-Explicit methods. In both cases, there was a significant increase in the length of the blade.
How to prevent the elongation of the generalized beam section in the Abaqus? It is noted that the natural frequencies of the blade should not change. For example, the amount of elastic modulus should not be increased.
Sincerely,
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Following
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If X is a quasi-Banach space without a separating dual that is no p-normable for some 0<p<1. Is it possible for there to exist a p-norm making X into a p-Banach space such that this new space is not isomorphic to X. In other words, are there non-trivial examples quasi-Banach envelopes.
If 0<p<1 then p-normability is equivalent to having Rademacher type p. So, if the ability to equip X with an p-norm implies X has type p in with its original quasi-norm then X is its own p-Banach envelope.
This question is uninteresting in the case X has a separating dual as such spaces may be equipped with r-norms for all 0<r<=1. And, it may be that many of these are not equivalent to X's given quasi-norm.
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I'm not sure of understanding the question. Nevertheless, in the case it was useful for someone, let me point put the following example: Let 0<p<q<1 and put X=l_p+L_q. The Banach envelope of X is l_1+0=l_1, which does not separate the points of X. The q-Banach envelope of X is l_q+L_q. So, X one-to-one embeds into its q-Banach envelope. In other words, the q-norm of l_q+L_q is a q-norm on X which is not equivalent to the p-norm X is equipped with.
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Hi everyone,
As we know, the formulas of the natural frequencies and their corresponding mode shapes for the uniform linear elastic soil (without damping) on the rigid bed are as follows [1]:
f_n = (2n - 1)vs / (4H)
vs = sqrt(G/ρ)
φ_n = sin((2n - 1)πz / (2H))
where f_n is the natural frequency of the corresponding mode in Hertz, φ_n is the mode shape of the corresponding mode, n is the mode number, H is the depth of the soil column, vs is shear wave speed, and z is the height from the rigid bed.
I create the finite element model (FEM) of soil using the commercial finite element (FE) program Abaqus/CAE 6.14-2 x64. The soil was modeled according to the geometric properties using the C3D8. It has a depth of over 50 meters and a horizontal distance of over 100 meters. The soil element size is 2 meters. The total model consisted of 62500 soil elements. Fig. 1 shows the 3D of the soil model. The unit weight of soil is taken 15 kN/m^3, Young’s modulus is 1000 kN/m^2, Poisson’s ratio is 0.35, Cohesion is 5 kN/m^2.
In order to closely match the theoretical solution, some assumptions have been made for the modeled soil. The soil rests on the rigid bed, modeled as boundary conditions restraining the bottom against translations and rotations at all directions. Further, the outer nodes of the model at the same level (Y equal) are tied using the MPC-Tie constraint to move together.
According to the theoretical formulas mentioned above, the first and second natural frequencies of soil column are obtained 0.0786 Hz and 0.2357 Hz, respectively. Nevertheless, the FE results do not conform to this order as the mode number. The first twenty-two natural frequencies of the soil FEM are shown in Fig. 2. The selected 3D mode shapes of the soil FEM are illustrated in Fig. 3. As can be seen in Fig. 1 and Fig. 2, the 1st frequency of the model occurs at mode numbers 1 and 2 and in two horizontal directions perpendicular to each other. This result is consistent with the theoretical results. However, the 2nd mode shape as the model occurs in modes 11 and 12, which is contrary to the theoretical results. Interestingly, the behavior of other mode shapes of the soil model is also strange.
Why do the order of the fundamental mode shapes and their frequencies of the soil FEM differ with theoretical values? Should further assumptions be made to fully simulate the FEM with the theoretical soil model?
The figures and the Abaqus/CAE FEM file are attached.
Thanks a lot in advance.
References
[1] Steven L. Kramer. Geotechnical Earthquake Engineering. Pearson Prentice Hall, 1 edition, 1996.
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I uploaded the input file (*.inp). It is compressed.
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I get two quaternions streamed from IMU-sensors. How do I calculate the (smallest) angle between them?
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The angle theta between two unit quaternions q1 and q2 (subscripts do NOT denote components) obeys
[sin(theta/2)*e cos(theta/2)] = q1*q2^(-1)
where e is the unit vector along the axis of the minimal rotation between the two quaternions and the asterisk on the right side denotes the quaternion product.
You always want theta to be less than pi and you don't really care about e, so you can compute theta from either
cos(theta/2) = absolute value of scalar part of the quaternion product q1*q2^(-1)
or sin(theta/2) = norm of the vector part of the quaternion product q1*q2^(-1)
Computing theta as the arc cosine is simpler because the scalar part of the quaternion product q1*q2^(-1) is just equal to the dot product of q1 and q2 regarded as four-component euclidean vectors. It's a poor choice, though, if the angle between the quaternions is small, because the scalar part of the quaternion product is close to unity in that case and the arc cosine is very sensitive to errors there. This, it's best to compute
theta = 2*asin(norm of the vector part of the quaternion product q1*q2^(-1))
I have no idea how the Cauchy distribution method (whatever that is) could be relevant.
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While optimizing machining parameters through GRA, in the final step GRG of initial, predicted and experimental parameters were compared to find the effect of GRA. In this regard on what basis the initial parameter combination selection has been made. In articles there is no explanation about this. 
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in several paper about grey relational analysis combined with taguchi methode , I found initial parameter design to compare predictive value in confirmation test . For example : initial factor A1B2C2D1 compare with optimal result A2B2C2D2. Anyone can explain how determine the initial parameter design or initial optimal faktor? I really confuse about this.... Thank you
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Is Entropy Shanon a good technique for weighting in Multi-Criteria Decision-Making?
As you know we use Entropy Shanon for weighting criteria in multi-criteria decision-making.
I think it is not a good technique for weighting in real world because:
It just uses decision matrix data.
If we add some new alternatives, weights change.
If we change period of time weights change.
For example we have 3 criteria: price, speed, safety
In several period of time weights of criteria vary
For example if our period of time is one month"
This month may price get 0.7 (speed=0.2, safety=0.1)
Next month may price get 0.1 (speed=0.6, safety=0.3)
It is against reality! What is your opinion?
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Once I was working on several variables and I wanted to weight them. At this time, people usually say that we'd better provide a questionnaire and then through AHP, ANP or other related methods define the weights for variables. That's quite common but how about the bias of the those who fill the questionnaire. Therefore, I looked for some other methods to weight variables based on the reality and I came across with Entropy. In fact, I weighted variables based on the each of these methods and then I compared the results. Entropy results were much closer to what is going on in real world.
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for linear control systems x_dot=Ax+Bu the reachability set can be calculated using the Image of the controllability matrix, i.e
R=([B AB A^2B,....,]) and reachability set=Im(R)
when rank(R)=n, and we do not have any control constraint the reachibilaty set of linear system is R^n (n is the dimension of states)
if we have a non-linear affine-control system
x_dot=f(x)+g(x)*u
R can be calculated using Lie algebra
R=[g1,g2,[f,g1],[f,g2],...]
my question is, in this case reachability set is again Im(R)?
and if reachability set=Im(R), how can we compute the rachability set, because here R will be a matrix with function arrays (function of states)
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I follow the answer
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I am simulating (CFX) a pipe with a turbulence promoter in it. I am using RNG k-e turbulence model yplus is lower than 1. For mesh independency analysis I am using turbulence kinetic energy as a output. The problem is, as I increase the mesh size, relative difference is getting higher up to certain mesh size (solution is getting mesh dependent). After that mesh size the relative difference is decreasing. For example from 4 to 12 million, relative change increases and from 12 to 25 million, it decreases. This values are coming because of the region where is just after the turbulence promoters (dominant anisotropic flow).
What could be the reason?
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Horimek Abderrahmane I will thank you very much
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One of the main stability theories for stochastic systems is stochastic Lyapanuv stability theory, it is the same as Lyapanuv theory for deterministic systems.
the main idea is that for the stochastic system:
dx=f(x)dt+g(x)dwt
the differential operator LV(infinitesimal generator- the derivative of the Lyapanuv function) be negative definite.
there is another assumption for this theory:
f(0)=g(0)=0
and this implies that at equilibrium point (here x_e=0) the disturbance vanishes automatically.
what I want to know is that is it a reasonable assumption?
i.e in engineering context, is it reasonable to assumed that the disturbance will vanish at the equilibrium point?
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From my practical experience if f(0)=0, then g(0) doesn't equal 0, because of the sensors noises.
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Imagine we have an ODE system
x_dot=[f1(x,u), f2(x,u), f3(x,u),....fn(x,u)]
where f1,..,fn are nonlinear functions of control input u and states x,
x is member of R^n and u is member of R^m
under which conditions we can change the dynamic of the system to arbitrary dynamic x_dot=[f1_des(x) f2_des(x),...,fn_des(x)]
where f1_des(x) f2_des(x),...,fn_des(x) are arbitrary pre-defined nonlinear functions.
is input to state controllability enough to do that?
Example
consider the 2-D system:
x1_dot=f1(x1,x2)+g1(x1,x2)*u1
x2_dot=f2(x1,x2)+g2(x1,x2)*u2
by choosing
u1=(f1_des(x1,x2)-f1(x1,x2))/g1(x1,x2)
u2=(f2_des(x1,x2)-f2(x1,x2))/g2(x1,x2)
we can change the dynamic of the system to
x1_dot=f1_des(x1,x2)
x2_dot=f2_des(x1,x2)
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Yes, it is. Controllability, or more precise input-to-state controllability, means that you can force all states behave as you desire, even for a single input.
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I'm studying the tensile behavior of a composite cylinder made of two different materials. The cylinders are concentric and placed one over the other. I want to assign an absolute stiffness to the contact elements between the outer surface of inner cylinder and inner surface of outer cylinder in Ansys. But as far I know one can only assign the normal stiffness factor in Ansys workbench.
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Hi,
The tangential stiffness can be assigned but not directly (Only by changing the normal stiffness). you can do the following in workbench (See attached):
  • Insert a contact tool under the connection branch and insert initial information
  • Select worksheet (RMB on any column and tick normal & tangential stiffness)
  • The values of Normal & tangential stiffness will be shown in new columns in the same worksheet.
  • keep changing the normal stiffness of the contact region and evaluate the initial information until you reach the required tangential stiffness.
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I'm teacher and suffering a lot to complete my MS. I need to write an MS level research thesis. I can work in Decision Making (Preference relations related research work), Artificial Intelligence, Semigroups or Γ-semigroups, Computing, Soft Computing, Soft Sets, MATLAB related project etc. Kindly help me. I would be much grateful to you for this. Thanks.
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The answers to the question for this thread are excellent. There is a bit more to add.
Before starting either a M.Sc. or Ph.D. thesis, it is very important to read published theses by others. Here are examples:
Source of M.Sc. and Ph.D. theses:
Another source of theses:
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in most cases for continuous time stochastic systems which are modeled by SDE, the Lyapunov stability conditions can guarantee the stochastic stability of the system,
another definition In stochastic literature is detailed balance which guarantee the convergence of the probability of the states of SDE to a stationary probability density.
I want to know which one is more strong stability condition?
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With Markov chains having a detailed balance distribution is a stronger condition then having a stationary distribution (that you can check by Lyapunov theory).
Every detailed balance distribution is stationary, but not all stationary distributions are detailed balanced. The proof is very easy. Detail balance means there exist a distribution pi(k) such that
pi(k)q_kj =pi(j)q_jk,
where q_jk is any non-diagonal entry of the generator matrix Q.
Now if on both sides you take the sum over all the k that are different from j you get that stationarity follows (pi Q=0).
You can check it on Durret's book Essentials of stochastic processes, page 130
I'm not sure what happens with diffusions, but it shuold be the same.
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For this analysis I like apply on a model  modeled in workbench.
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Yes huseyin. Try SMART CRACK GROWTH function in workbench. You may see lecture provided on youtube.com
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First a large size matrix requires sufficient memory to inverse the matrix. Secondly, there are several mathematical techniques are available to solve the inverse of a matrix. But in handling a large matrix, still I couldn't find any faster and accurate method which can solve this problem with less memory consumption as well.
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It depends on the matrix and no universally fast method exists!
In particular its eigenvalue characteristics and rank of the matrix or the pattern of non-zero elements. For sparse and patterned matrix which are usually seen in numerical solutions of PDEs (like FEM and FDM) there are well established methods which are developed over years and some of them are very efficient:
In general, if you want to invert a full matrix of size N X N you have to do O(n3) arithmetic operations (without applying any numerical tricks).
But we have methods for inverting sparse matrix of size N X N which are as efficient as O(N log N) like Thomas Algorithm (see below)
If you want more in depth discussion on numerical method s for inverting a matrix, there numerical efficiency and palatalization see these three:
1- Yousef Saad web page:
2- Works by Roland Freund (UC Dvais Department of Math)
3- Book by Gene H. Golub of Stanford:
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In optimization problem often we use local optimum but is it global? Or are there any meta-heuristics algorithm to obtain global solution? If there any then what is the name of that algorithm and if possible how we can get that solution?
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I wish not to let you down, but the basic answer is "very seldom", and another one is "you will not know if you have stumbled upon an optimal solution, because there is no natural termination criterion based on the concept of optimality". (In contrast, a branch-and-bound, or branch-and-cut, methodology is based on local AND global bounds on the optimal value generated throughout the procedure, and in most cases the correct procedures will either fix some variables to their optimal values before termination, and they will be able to discard a very large portion of the search space based on parts of the search space being infeasibie or inferior, in which case we do know for sure that an optimum has been reached.)
If you have a structure of the problem that makes it emanable to be solved by special methods, such as Benders decomposition - when you have a mix of integer variables and continuous variables. you also have a fail-proof method.
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There are lots of Optimization method /Evolutionary algorithms (EAs) in literature. Some of them is more effective (for solving linear/nonlinear problem) compared to other. But we don’t know which will fit our model. As a result we checked for everything as we can do. But cant get the desire result. Some of those methods are 1. Genetic algorithms (GA) ; Haupt and Haupt (2004) 2. Pattern search (Mathlab) 3. Particle swarm optimization (PSO), Binary Particle Swarm Optimization (BPSO); Eberhart and Kennedy (1995) 4. Bee optimization; Karaboga and Bosturk (2007) Pham et al (2006) 5. Cuckoo algorithm; Yang and Deb (2009, 2010) 6. Differential evolution (DE) ; Storn and Price (1995, 1997) 7. Firefly optimization; Yang (2010) 8. Bacterial foraging optimization; Kim, Abraham and Cho (2007) 9. Ant colony optimization (ACO) ; I Dorigo and Stutzle (2004) 10. Fish optimization; Huang and Zhou (2008) 11.Raindrop optimization ; Shah-Hosseini (2009) 12.Simulated annealing ; Kirkpatrick, Gelatt and Vecchi (1983) 13.Biogeography-based optimization (BBO), 14. Chemical reaction optimization (CRO) 15. A group search optimizer (GSO), 16. Imperialist algorithm 17. Swine flow Optimization Algorithm. 18. Teaching Learning Based Optimization (TLBO) 19. Bayesian Optimization Algorithms (BOA) 20. Population-based incremental learning (PBIL) 21. Evolution strategy with covariance matrix adaptation (CMA-ES) 22. Charged system search Optimization Algorithm 23. Continuous scatter search (CSS) Optimization Algorithm 24. Tabu search Continuous Optimization 25. Evolutionary programming 26. League championship algorithm 27. Harmony search Optimization algorithm 28. Gravitational search algorithm Optimization 29. Evolution strategies Optimization 30. Firework algorithm, Ying Tan, 2010 31. Big-bang big-crunch Optimization algorithm, OK Erol, 2006 32. Artificial bee colony optimization (ABC), Karaboga,2005 33. Backtracking Search Optimization algorithm (BSA) 34. Differential Search Algorithm (DSA) (A modernized particle swarm optimization algorithm) 35. Hybrid Particle Swarm Optimization and Gravitational Search Algorithm (PSOGSA) 36. Multi-objective bat algorithm(MOBA) Binary Bat Algorithm (BBA) 37. Flower Pollination Algorithm 38. The Wind Driven Optimization (WDO) algorithm 39. Grey Wolf Optimizer (GWO) 40. Generative Algorithms 41. Hybrid Differential Evolution Algorithm With Adaptive Crossover Mechanism 42.Lloyd's Algorithm 43.One Rank Cuckoo Search (ORCS) algorithm: An improved cuckoo search optimization algorithm 44. Huffman Algorithm 45. Active-Set Algorithm (ASA) 46. Random Search Algorithm 47. Alternating Conditional Expectation algorithm (ACE) 48. Normalized Normal Constraint (NNC) algorithm 49. Artificial immune system optimization; Cutello and Nicosia (2002) 50. fmincon .
Besides this there are many other optimization algorithm recently invented which are generally called Hybrid optimization Technique because it’s a combination of two method. If we share our experiences then it will be helpful for all of us who are in the field of optimization. I may be missing some methods, researcher are requested to add those algorithms and the way of use like many model needs initial value, weight, velocity, different type of writing objective function etc. I am facing some problems that’s why I make this format which will definitely help me as well as all other researchers in this field. Expecting resourceful and cordial cooperation.
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Abu, you are not very clear. If you want to discuss effectiveness, then you need to define what that is, and run experiments on a huge set of carefully randomized test problems in order to make any claims about one method being better than another. I have mentioned on more than one occasion that if you have a deep knowledge of your problem and its properties, chances are very slim that a metaheuristic will be the winner against your own devise.
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Is it possible to manage with supply chain in a more effective way?
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Dear Abu Hashan Md Mashud ,
Kindly clarify your question.
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Like wave propagation in magneto-thermoelastic media. In Finite element formulation I found the second derivative. I am little bit confuse about continuity of the second derivative for linear element.
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I had the same question, and I found this equation.
I think this could be helpful
Regards
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I have two stationery time series that I want to see are they correlated or not. I decided to work with cross correlation, there is a good answer in Correlation between two time series but I dont know how to calculate p-value(as http://stats.stackexchange.com/users/11032/michael-chernick said) and decide that its a good answer, if there are correlated do something. I dont know how to do this. I want to code it with C++ and use it for simulation in Omnet++ . Could someone help me to find a way or introduce a document to me to understand the mathematical way to calculate it and program it in C++? P.s : My data are the number of packets which send from two nodes to an OLT in a network continuously. something like this:
from time 0 to 10:
series 1 : 2 5 3 7 9 2 0 1 8 4
series 2: 3 4 6 0 5 9 2 5 3 7 . series 2 has 6 min delay. or maybe there is no delay in other case, I must calculate the delay also (I can calculate delay with omnet++ but I dont know are they correlated at the end or not)
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Did you figure out how to calculate the p value for cross-correlations?
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I have an image with a bigger scale in Y than in X. In this image I want to measure several areas. Now I need to know, if the true area is dependent of the orientation of the object / area I want to measure.
Look at my example:
A is the original picture. All bars are of the same size and area.
B is picture A with a bigger height (Y) scaling
Do all three bars (horizontally vs. vertically vs. diagonally) in B have the same area?
So to summarize my question again: A picture with objects of the same size gets changed in Y-scale. In this changed picture, do all objects still have the exact same size?
Thank you very much!
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Let A=(a1, a2) and B=(b1, b2) are vectors forming two neighbor sides of a parallelogram ( rectangular in the question). Its square is
S = | A x B | = | a1 b2 - a2 b1 |.
The transformation
x1 = k1 x
y1 = k2 y
translates A and B to
A1 = (k1 a1, k2 a2),
B1= (k1 b1, k2 b2).
(In the question k1=1).
New square
S1 = | A1 x B1 | = | k1 a1 k2 b2 - k1 a2 k2 b1 | = | k1 k2 | S.
Thus, all initially equal squares remain equal.
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What are the applications of Diagonalization of a matrix?
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The main purpose of diagonalization is determination of functions of a matrix.
If P⁻¹AP = D, where D is a diagonal matrix, then it is known that the entries of D are the eigen values of matrix A and P is the matrix of eigen vectors of A.
If the matrix A is invertible then A = PDP⁻¹
Then f(A) = P*f(D)*P⁻¹
This expression is very useful because f(D) is simply a diagonal matrix whose diagonal elements are f(D1), f(D2)........f(Dn) where D1, D2.......Dn are the eigen values of matrix A.
f can be any function such as sin, cosh, sqrt ...... etc. In essence diagonalization provides a simple pathway for evaluation functions of a matrix.
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Model Predictive Control is a receding control approach, that basically does online dynamic optmisation on a finite horizon while implementing online, the first control optimal action of the control sequence. The online dynamic optimisation is repeated based on the measure system states with a sliding finite-horizon window.
However, can't we say that in principle due to the finite-horizon nature of MPC, the solution result obtains does not guarantee global optimality in the infinite horizon time scale as the control sequence found between A-B will not be necessarily optimal if the horizon A to C (A-B-C) was considered ? It seems to be some kind of corrective greedy search where the best control sequence at any branch is selected and because the control sequence is not fully implemented at the finite horizon, there is a possibility for escaping a bit the danger of the greedy search at the next iteration?
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Dear Yves,
Because in the infinite horizon MPC the resulting problem is an infinite dimensional problem that in most cases is computationally intractable and thus the true "Optimal" solution can only be approximated with the discrete time finite horizon equivalent and under assumptions about the origin of the system and its active sets. There is a very good research in this paper that you might want to look at: Pannocchia, Gabriele, Stephen J. Wright, and James B. Rawlings. "Existence and computation of infinite horizon model predictive control with active steady-state input constraints." IEEE Transactions on Automatic Control 48.6 (2003): 1002-1006.
Best,
vassilis
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Given an undirected(weighted) graph depicted in the attached diagram which is a representation of relationships between different kinds of drink, how best can I assign weight to the edges? Is there a technique that I can use?
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Use Structural Equation Modeling (SEM).
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I am a researcher of materials science and metallurgical engineering. I am interested to work with atomic level and electronic modeling of the structure and properties of materials,materials informatics and simulation in future. Currently i am learning Python. I have basics of C and MATLAB . What will be the useful programming languages to learn for me?
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Hi Sajib,
it depends on your needs. Are you planning to use some commercial softwares? For example, for Ansys you need to use the ANSYS Mechanical APDL language. In Abacus you are can program your own material model in Fortran. In COMSOL Multiphysics you can connect to Matlab via LiveLink and so long :). So find out, if you are gonna use some special commercial software and try to focus on the language needed therefor.
Best regards,
Antoni Artinov
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What I want
____________
Circulation = path integral of the velocity around the boundary of surface --> (Stoke's Theorem)--> Circulation= surface integral of curl of the velocity field over the surface
What I have
____________
* A number of N points defined by their coordinates (x, y,z). These points are on the surface (not the contour of the surface)
* velocity field components u,v,w for these points
* Form (visually observable) of the surface. Is not rectangular. It is more like a circumference.
(I'm using Matlab)
Thanks in advance.
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OpenPIV Spatial Toolbox does vorticity from PIV maps, the equations are visible in the code itself. Disclaimer - I'm one of the developers.
PIVMAT also has this option, http://www.fast.u-psud.fr/pivmat/
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We can think of the circle center as a "focus" of points of its circumference given by distance to center = constant. In case of an ellipse we have sum of distance from points on boundary to the two foci =constant, Can this be generalized ? In fact the reverse problem is of interest i.e. given an arbitrary closed curve, is it possible to determine such criteria( or numerical algorithm to identify such foci and also conditions when it exists). This is a general exploratory question. and all are welcome to comment.
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Actually, not. There is no common rule to determine a focus of a curve (beyond the second order ones). Of course you can determine infinity of "foci" by different laws. For example, a set of points such that d1^2 + d2^2 - d1*d2 = const. However you ask about inverse problem, so my answer is - will not.
P.S. Naturally, a focus is associated with the whole curve, and if you
transform a piece of circle (with homeomorphism), its focus will be broken. Instead of term "focus" you may take its local analog - center of curvature.
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Often computational time estimation for a numerical operation is based on analytic data. And the sparse matrices are often considered in engineering structural analysis. Thus, the operation counts of matrix analysis (including matrix multiply, matrix decomposition, and solution of linear systems) are often needed to be known in a analytic way. Therefore, anyone who would like to suggest some literatures and opinion are encouraged to discuss them to me.
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Most processors can do an addition, comparison or multiplication in a single cycle : those are all counted as one Flop. But division always takes longer (4 Flops in the HPC community). Flops may vary by CPU. You must know your hardware to compute the ideal Flops on the given architecture and you have to know the implementation of your algorithm it is a useful tool which tell you how you utilize the CPU. An 80387 math coprocessor took something like 300 cycles for a single multiplication. Most recently any modern intel core can perform about 2 Flops per cycle. A recent CPU will easily achieve several Giga-Flops (billion floating point operations per second). An i7 processor can perform 150 Giga-Flops. If we take exactly one second for an operation such as adding, substracting or multiplying two values the performance is simply 1 Flops (i.e 10^9 operations = 10^9 Flops). Measuring the performance of a program needs to count the number of operations. Several researchers have considered the For loop as an instruction but its real implementation takes an addition i++ and a logical test i<n (comparison) at each iteration for example in classic algorithms or in sparse ones: we have then n additions and n tests. You have also to consider the processes per second for putting data in memory and retrieving them from memory (scanf) other than calculations. Attributing data to variables doesn't take much time.
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- Could you please point me out to some successful Medical sciences applications using partial differential equations?
- Preferably, involving heat, reaction-diffusion, Poisson, or Wave equation.
- If possible in fuzzy environment.
Best regards
Sarmad.
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Dear Professor, In mathematical modelling for drug delivery we are using PDE.
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What is the fastest way to learn about neural networks? there types? how to use them? full solved examples? real life applications?
Many thanks
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Coursera
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Could you please point me out to some successful Signal, image, or video processing real life applications using partial differential equation?
Preferably, involving heat, reaction-diffusion, Poisson, or Wave equation.
If possible in fuzzy environment.
Best regards
Sarmad.
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We are trying to simulate mixing of two streams of different chemical concentrations.
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Yes Ahmad Faris Ismail Sir, there are many multiphase fluids options in Comsol.
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Mathematics is fundamental in all sciences and mainly in Physics, which has even had many contributions. It seems that the capacity to be applied would be the motor to be create. But this not what good mathematicians as Henry Poincarè or Hardy has said. What is the beauty in mathematics, in theoretical physics or in others which could be related subjects?
For me there are very beautiful mathematical results which sounds difficut to be applied or even against our reality, which are full of "beauty" or at least "surprise".
1.Sum of natural numbers = a negative number as - 1/12.
2. Polynomials with degree five or higher are without analytical expression of their roots.
3. Banach-Tarsky theorem
4. There cannot exit more than five regular polyhedra in three dimensions.
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The beauty of theoretical physics is that Maths is it’s language. The beauty of mathematics is in its remarkable success of describing the natural world.
it is therefore not surprising that most research mathematicians and theoretical physicists pepper their description of important research work with terms like “unexpected,” “elegance,” “simplicity” and “beauty.”
Let me make it easy for you:
Can you imagine a bride without a wedding dress?
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Which analysis could use for this problem?
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hi Iarly Vanderlei da Silveira
please call me in: seyed.reza.991@gmail.com
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currently some colleagues and I have been working on these issues and the curious thing is that we do not find articles or works that are not more than 10 years, it is as if the enthusiasm for this subject would have died.
thanks for any help.
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What exactly you are interested in?
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Some of students ask me to give the applications of mathematics in real life. What are some of the interesting applications of mathematics in real life? Could you please share your knowledge about this issue.
Thank you very much in advance for your cooperation.
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You are asking about millions of mathematical application in our life, like calculation of area, volume, velocity, acceleration ....etc.
Regards, Emad
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I want to study the flow pass two squirmers, which corresponds to solve the Stokes equations for two spherical bodies (some distance apart) with prescribed tangential velocity on their surfaces, in an otherwise static fluid. (You can find these equations in the attachment)
We can find analytical solution when there is only one squirmer at play, but exact solution become hopeless for two squirmers. Due to the linearity of the equation, I once considered the superposition of two one-squirmer solutions (After some shift of the coordinate), but this would break the boundary condition and make the squirmer unrealistic.
So now I am looking for the solution of aforementioned equations numerically, I also restrict myself to 2-dimension. But it is still hard to express the boundary condition on two circles simultaneously (the polar coordinate won't work). My plan is to find another coordinate system that transforms these two circles into two lines at infinity, then try to solve it with Mathematica or Matlab (not sure whether it will work).
Could you please share your thought on this problem? Is there any other method/software I am unaware of that can crack it easily?
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Dear Yuanzhao, I suggest you to try a symmetry method using the central stright line between the two spherical bodies as axis of symmetry; so you can use the center of a body as origin of polar coordinates. I think this method should work fine.  Gianluca
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How can I compare the shape of two curves? I was reading about techniques like the Fréchet distance, but I do not think that is useful, because some curves look alike, but are for example positioned more to the left on the X-as. How can I compare only the shape, not the position on the X-as? Thanks in advance.
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-Quientens
My suggestion is that select few points find the slope .Then you will able solve your question.
Biswanath
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I wish to learn meshfree methods. Please recommend some good tutorials and example codes.
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Hi.
Maybe you can consult the follwoing two books. I did not get any hits with online.
(1) Meshfree Approximation Methods with MATLAB
(2) Advances in Meshfree Techniques
They were published around 10 years ago. Hope they are helpful for you.
All the best.
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Dear All,
I have a question regarding the the existence and uniqueness of a system of linear Volterra integral equations.
Does the solution exist? Is it unique?
What are the conditions on the kernels for the existence and uniqueness of this system of equations?
The equation is written in the attached file.
I would appreciate your help greatly. Thank you.
Sarah
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From practical perspective I feel that the uniqueness of the solution would depend on how many parameters you have in the set of equations.  In one case, I had 5 parameters.  It is like trying to fit in a 5 dimensional space.  Such solutions cannot be unique.  However, they do shed light on various other aspects and that itself is valuable.
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Dear all,
I have developed a new technique which solves linear integro-differential equations of fractional type. This includes Fredholm and Volterra equations.
I am looking for an application which can be modeled into such equation so I can apply my method. It can be any kind of application.
I also solve mixed system of equations e.g.1. A system of multiple Fredholm equations of different order of fractional derivatives (0, 1/2, 1, 3/2, etc..) or e.g. 2. A system of same or different kinds of Volterra equations. So, if there is an application to this kind of equation, it would be great!
I would appreciate your help. Please refer me to an article.
P.S. Only linear equations please.
Thank you very much in advance.
Sarah
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Dear Sara,
First of all, let me congratulate you for your achievement.
Do not bother about applications. Go on your research,
But if you want to have an application, you can see
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As stated in the documentation of the "distributed parameter line" block: "This model does not represent accurately the frequency dependence of RLC parameters of real power lines. Indeed, because of the skin effects in the conductors and ground, the R and L matrices exhibit strong frequency dependence, causing an attenuation of the high frequencies."
Is there a third party model of transmission line, compatible with SimPowerSystems, dealing with skin effect and other nonlinear behavior?
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Hi, i search Matlab code for plotting the transfer function of a channel using ABCD parameters with a network of three branches in the frequency band of 3-500KHz.
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I need to model a DC load, that is connected to a 3-terminal DC grid. The other 2 terminals provide the needed power. As far as I know, the load can be modeled as a pure resistance, but I don't know how its value should be determined or if there are better and more accurate ways to model it.
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As Prof. Briz correctly says, CPL as a resistance is only accurate for the very low frequency region (DC). At higher frequencies, the active device (i.e. the power converter) gradually loses its control action/regulation ability and then, the equivalent impedance tends to be dominated by the  passive components (e.g. capacitances).
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For analysis of engineering structures like geometrically nonlinear beams and plates in statics, I consider two main approaches: finite element method (FEM) and finite difference method (FDM). Since I am just starting, could you please explain me:
1. if I use FEM, is it possible to obtain a discretized system of the form Ax=b, where x is the displacement of the structure? As I understand, FEM reduces the problem to Kx=f, where K is the stiffness matrix and x is the coefficients of the approximation of the displacement with ansatz function. Am I right?
2. What are the advantages of FDM over FEM especially for nonlinear problems?
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I have amended the link, which should now work.
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Currently I am working on Computational Fluid Dynamics and Heat Transfer study of a mixing T junction. I have carried out analysis and have obtained results for the same.
My query is I want to convert the model made into 1 part 1 body instead of 1 part 2 body so that the grid generation becomes feasible and smooth.
Please guide me for the same by describing the steps against how to overcome the problem.
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sorry design modeller and mesh module in ANSYS 17.1
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I am conducting a survey of analytical methods, from engineering cycle analysis to 3D full N-S coupling detonation, turbulence, multi-specie fluids, codes having the capability to assess propulsive performance, and to model/simulate computationally any detonation-based propulsion systems such as PDEs and RDEs.
In addition to the capability of the tool, I'd like to know if the code is proprietary, ITAR, commercially available, or if it can be obtained for academic research.
Thank you!
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Well on a mcro scale all I.C engines are detonation engines. If you one can create detonation at normal temperature and pressure that  could be possible with chemical reaction between two fluids of different surface tension.
As acild used to do a simple experiment. take a small leaf of grass and coat the large end with soap and float in water and the grass leaf will jerk forard like a boat.  the movement was intermitent.
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I want to know how to calculate the Gibbs free energy in VASP . I have 1 T phase 2D materials structures and I want to know which kind of parameters I need to fixed in the INCAR file through I can get it. Where I will find these energies in VASP calculations? Please some one guide me about this.
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Total energy at 0K is the Gibbs free energy. You further understand it through Gibbs free energy equation, as other terms have no meaning at 0K.
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Hi,
  In FEM, I imposed the Dirichlet as essential and Neumann as natural boundary conditions for the heat equation with some specific values. My question is : Once the solution is obtained for the temperature distribution, how to know whether the boundary condition is satisfied or not? As far as Dirichlet boundary condition is concerned, it satisfies exactly with respect to the domain since it is explicitly applied and in the solution, the values are same. What about Neumann Boundary condition?
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@Amaranath
Simple way is to plot the solution. Then you will know your requirement.
B.Rath
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I need to know if there are some emperical equations or any kind of formulations that can be used to model and analyse masonry structure as a nonlinear anisotropic  and homogeneous material.
what are the parameters( mechanical properties of the masonry) needed to do so?
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Theقث is now a well engineering theory on masonry structures
I am now teaching masonry structures for master degree students in civil engineering
Masonry is classified as plane and reinforced
plane masonry are (according to Masonry Specifications Joint Committee (MSJC-11))
 are analysed within the linear stress strain relationship range because its strength is governed by tensile strength ( in case of panel wall subjected to out of plane loading)
on the other hand the reinforced masonry uses similar relations and equations as those used in reinforced concrete
the most important parameters in masonry are
1. modulus of rupture ( depends on masonry units and type of mortar or grout)
2. compression strength of masonry ( a function of unit compression strength and type of mortar)
3. Shear strength 
4. modulus of elasticity ( = 900 and 700 times compression strength of concrete and clay units respectively)
the compression strength is determined according to ASTM C 1314 or by using  tables 1 and 2 of MSJC-11   
you can find the required information in the following references:
1.James E. Amrhein and Max L. Porter “ Reinforced Masonry Handbook” 6th edition 2009, MIA & ICC
2.Richard E. Klingner “ Masonry structural design” McGraw-Hill 2010.
Narendra Taly “Design of reinforced masonry structures” second ed. McGraw-Hill 2010.
3. TMS 402-11/ACI 530-11/ASCE 5-11 “ Building code requirements and specifications for masonry structures”, MSJC 11
4. IBC – 12” International building code”
5. ASCE7 – 10 “Minimum Design Loads for Buildings and Other Structures”
in addition you will find one of my lecture on masonry in the attached file
As you need any help, please feel free to send me a message
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hi
How find centroids vertices using LOF function in highlighted section(Definition 3) in attached file ?
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The first letter  -- a --  is a misprint and should be removed.
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If possible, I needed comparison in solver capabilities, material database, accuracy of simulation output, licensing cost etc. What software would be recommended for specific Sheet Metal Forming Simulations?
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You can go for PAM STAMP which gives good results. It include a CATIA die maker module to design die faces, both implicit and explicit simulations can be done with it.
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I have developed a system dynamic model containing approximately 66 parameters. I have a reviewer who thinks the model is too simple as I could have added more inputs. However, my decision to exclude such inputs was based on the fact they do not provide any significant information about the performance of the model. 
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A complex model constitutes the mathematical description of a complex object, the one that consists of interrelated component elements, that can also be constituted by their own interrelated elements. Also, the complex object and their elements have functions and also can be break down in the time, because in the general case it is required making decisions related with the system in different time periods. On the system and their elements act interferences of the surrounding world. While more complete will be the organization of the system smallest will be the interferences, but higher the complexity of the conciliation of it operation.
The reading of the work that I am attaching could help you to understand what a complex object is, what is a complex model is and how to organize the decisions making processes considering this complexity.
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I want to try out numerical modelling of helical soil nails.
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hello dear Saurabh
i have a paper in this area and when we modeled the helical piles in 2D the results was not good and we had to modeled it in 3D abaqus
because some loading type have not good result and matching with real results
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I have applied torque to the beam and want to see the deflection, not deformation. But, I have seen  only deformation in the solution list. So, can anyone help me to find the deflection of the beam in ansys? 
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You can get the deflection by using general post processor (you can see the contour  with colour bar) and time-history post processor (you can get the deflection data at nodes: Select TimeHist Postpro > Variable Viewer -----then----Nodal Solution > DOF Solution > Y-Component of displacement. Click OK). If you want any other help, feel free to message me.
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There is a multitude of well balanced schemes for the shallow water models with source terms.
Which five of them can you recommend? The hydrostatic reconstruction is actually top on my list. So what do you think?
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There has been great emphasis in the literature on the derivation of well balanced schemes for the shallow water (SHW) and similar equations in the case in which both the continuity equation and the momentum equation are in full flux form. However, for many applications (especially in subcritical regimes), in my opinion,  there is no real necessity of employing the momentum equation in full flux form, since strict momentum conservation is often not really necessary (especially when dissipative terms are important and, as often happens, uncertain). In these cases the pressure gradient term is best written in non conservative form (see e.g. what is done in this paper
Rosatti, G., Bonaventura, L., Deponti, A., & Garegnani, G. (2011). An accurate and efficient semi‐implicit method for section‐averaged free‐surface flow modelling. International Journal for Numerical Methods in Fluids, 65(4), 448-473,
but there are many others also doing this). If you do that, the well balancing property is essentially automatic and granted for free and you do not need to worry about which approach to choose. Going deeper into well balancing issues is only really necessary when strict momentum conservation is essential (for example, when solving dam break problems). However, you can see from the previous paper that good solution for dam break problems can also be achieved for SHW equations formulated as suggested above.
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I need to optimize a function of several variables moving in R, under the costraint that only 2 of these variables can have a contemporary deviation with respect to the nominal value (In other words, N parameter variations must be explored, but only 2 at the same time, while others N-2 keep a known value). Do you have any suggestion about a method, or a paper which faces this problem using GA or PSO. Thanks.
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Dear Antoninio,
At a first glance I would say that you have to build n*(n-1)/2 problems by considering for each problem only two free variables. Then you can solve each problem by using a classical GA or PSO method.
Richard
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Well, it is about underwater vehicles moving on the concentric circles for tracking. Let the vehicles be XA on circle A, and XB on circle B, XC on vehicle C and so on. (There could be more than one vehicle on each concentric circle).
Since the circumference on A is smaller than that of the others, the vehicles will finish the tracking on A faster. After finishing the tracking, they will move to B to assist in finishing the tracking the vehicles on B have not yet finished.
When the vehicles XA and XB now finish the tracking on B, they will also all move to the next circle to assist in the remaining tracking the the vehicles there have not yet finished. This will move on until the last circle is tracked.
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It may be suspected that your goal is to simulate the described process by computer program.  If so, then you need some kind of equation saying what is the evolution rule, i.e. expressing the state of your system at time tn+1 as a function of its state at time tn (previous one).  But what is the state?  It is an array with as many rows as there are vehicles and 3 columns.  First column says at which track the vehicle currently is, the second may indicate its current position (may be expressed as an angle of rotation), and the last column indicates whether the vehicle is on the still unexplored part of circle or - perhaps - in a transit state, i.e. on its way to the next circle (this takes some time, isn't it?).
The most troublesome part is keeping track of the current degree of exploration of each circle.  You have also to decide whether the vehicle has to switch to another circle after reaching already explored part of current circle or later.  The other question is what to do when right after switching to the next track we happen to be on its already explored part ?
To perform such simulation we we need the complete description of an initial state, of course.
Hope this helps.
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I am trying to do some finite element dynamic model of the foot and ankle during running.  One of the critical model inputs in my thoughts is the magnitude of foot extrinsic muscle forces during running.  I have seen the same data for barefoot walking.  However, till now I couldn't find any for running.  Is there a data source or reference for the above mentioned data?  Thanks in advance. 
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i don't know
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I have 450000 data from gyroscope and I want to model stochastic noises of it and at bias instability modeling I need time correlation to model bias instability and I don't know who can I compute it?
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Dear Amir Taha,
see the formula for  time correlation in the file. Here R(t) is autocorrelation function.
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I am doing a very simple simulation in COMSOL consisting of piezo, matching layer and water. It calculates transmission coefficients for a range of frequencies given a constant matching thickness. The maximum of the curve matches with the analytical result ( Which I am pretty sure is right since it matches with the experiment) but the minimum values differ by 50%. i don't know why it is happening? Can anybody help me with that?
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Hi,
It's difficult to answer without seing the exact problem, maybe you could post at least your non-matching curves and some informations on the layer and piezo dimensions ?
As for the problem, remember that piezos are assumed lossless by default, and the same holds for pressure waves. Does your analytical result account for dissipation ?
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I calculated the band structure of O2 adsorbing in CH3NH3PbI3. I got the result as the following attach. who can tell me why it appear the fat band. anyone has suggestion?
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Dear Sophia,
Adding to Viorel who gave a satisfactory answer, the energy band structure of pervoskite crystal should not contain any allowed energy levels in the energy gap. However, if the material is doped with low density impurities, allowed energy levels appear inside the energy gap. The position of the allowed state depends on the energy needed to transfer an electron from the impurity to the valence band. This is very known phenomena and is exploited for doping process.
Also, if you interrupt the periodicity of the crystal at its surface without adding any any impurities at that surface, also allowed energy state will appear near the middle of the energy gap. The density of such state can be varied by passivatation with atomic hydrogen or oxygen. But in this case it is a chemical rather than physical absorption.
Physical absorption may not affect appreciably the electronic states.
You can your self study all these cases.
wish you success
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Hi all, I'w working with meshing using VisCART of ESI-Group. I have this problem: 32GB of RAM is not enough for my complex geometry. I set the file paging dimension to 80GB on the SSD drive. So now the system has almost 110GB of memory available. Anyway VisCART crashes without reaching all the available memory: I'm watching the 'Commit' tab of the Memory on the monitoring resource display of Win10, and never it reach 80GB; it crashes when it reaches about 55GB. Does it depends on VisCART? Or Windows? Anyway is it useful to do this for very complex meshes? Is it possible that a software is not able to manage all that memory?
Is there a free tool to optimize the performance of the memory utilization? Is there a method to understand the problem? For example a tool that gets all memory status when a process crashes?
Is it true that this operation is dangerous for the SSD drives? Is better to use an old hard disk?
Thanks
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Saverio:
By using a Linux distro you get access to both a high performance operating system,
a vast amount of free and open source software for science/engineering, and a community of dedicated software developers eager to help you.  You may also find a free software that can do what you are trying to achieve with VisCart. For example, Paraview and Visit are great open source packages used for scientific visualization. As you develop your skills, having a good working knowledge of Linux will also allow you to easily transition to clusters used for high performance computing (HPC).
Ubuntu is just one of many Linux distributions. A distribution (distro) is a tailored set of software packages, a package manager application, and a support community of developers/users.  Ubuntu is commonly used to help Windows users ease the transition to Linux, but it will most likely not be the final distribution you will end up using the most. For example, the distributions I have used and favor are debian, RHEL/Centos, and Gentoo. Ubuntu is a derivative of Debian.  There are many other distros, each tailored to a particular purpose. Regardless, switching to any Linux distro will likely make you much more productive.
As I mentioned in my first post, if you really need to use the VisCart software, I would contact the organization and ask what distribution their software is guaranteed to work on.
While you can make a fixed dedicated swap partition during installation, a swap file is more versatile and very easy to manipulate from any distro's command line. Simply create a file of the size you want using "dd", then format it to be a swap file with "mkswap", then activate it with the "swapon" command. Regardless, the swap file approach is not going to give you the performance you need, only increasing the available RAM will do that. However, using a Linux distro may avoid having a system crash after 55GB of swap is allocated. Although, it could be crashing for other reasons, such as a hardware problem, or a bug in VisCart.
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Please give some input if u can.
I m getting higher value of natural frequency of shell structure following TSDT than that with FSDT using Finite element method. It should be the other way. I double checked my code written in MATLAB but could not find any fault or error. 
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Here are a few things I would check:
1) Make sure that you have benchmarked your code against known analytical solutions for simple cases, such as a clamped or simply supported flat plate/cylindrical shell.
2) If your code accurately predicts the analytical solutions then vary the mesh size and perform a convergence study for your current shell geometry. 
3) Take a look at the mode shapes corresponding to each natural frequency for clues.
4) How much difference is there between the two results? Your code might be suffering from numerical round-off or floating point rounding errors. Consider the precision of your routines.
5) What is the thickness of your shell element? If the elements are not thin enough you could be in a regime where both theories perform poorly and are inaccurate. In the case of analytical solutions, shell theories typically require a length to thickness ratio of greater than 20. (L/H > 20)
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How can calculate fracture load of three-point bending testing through strength and ductility data obtained from tensile testing?
and in three-point bending testing ,how to calculate stress distribution of crosssection beyond elasticity scope?
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According to the current thinking you cannot do this, at least not universally, i.e. across all materials. Although there should be a link between tensile properties and fracture toughness, for the moment the toughness is considered a separate material property that needs to be measured, e.g. by 3PB specimen. Hence you need to measure the fracture load of such a specimen, not infer it from tensile test stress-strain curve.
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A q-ary necklace of length n is an equivalence class of q-coloured strings of length n under rotation. The number is
Nq(n)=(1/n) \sum_{i|n}\varphi(n/i) q^i
For example q=3, n=3,  N3(3)=11
000 000 000
001 100 010
002 200 020
011 101 110
012 201 120
021 102 210
022 202 220
112 211  121
122 212 221
111 111 111
222 222 222
But if the first bit must take modulo 2, then the answer is 4. For example
001
002
011
122
Here 210=010, 001=201 , 000=200 and so on. Note: the first bit must take modulo 2.
What is the number of Necklaces in length n and beading of q different colors with the first bit taking modulo 2?
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I do not know, whether it will be helpful for you, but I proposed some method. In any way, I would be glad to hear any comments.
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Dear Hedieh Alipour
I enclose for you one of the  M. Sc. Thesis that covers the  design of Many circuits using Memristor.
Good Lick
Prof. S. EL-Rabaie
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The collocation method is one of expansion methods which is used to approximate the linear and non-linear integral equations. This method can be used with basis functions such as standard polynomials {1,x,x^2,x^3,...},  Taylor polynomials, Chebyshev polynomials (first, second, third and fourth kinds) and others. What are the orders of convergence of collocation method with standard polynomials and Chebyshev polynomials ( second, third and fourth kinds)?
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May be useful this expression try this Order of Convergence is equal to [log(max(error(N/2))/max(error(N)))]/log(2)