Can anyone suggest any software that will calculate increment stress effects on a body ?
a particular application would be calculating increment stress on gear teeth through a simulation run.
Since we would have a cyclic run, if we had 2 gears, their teeth would be in contact once every revolution, and i am interested in knowing if there is a software that will keep track of the "damage" done on first contact, which would slightly change the geometry of the gear and most importantly change the way the gear responds to the same stress at future contacts.
You'll need a non-linear transient FEA capability.
I'm assuming that the gear rotational velocity is small enough where you aren't interested in inertial effects. You want to do a non-linear load that tracks loading over one or more rotational cycles.
You need to model contact and friction at the contact points. That's a challenging non-linear problem.
You'll need a mesh that's refined enough in the contact zone to resolve the surface stress you're interested in.
Small strain is sufficient as a first step. Large strains would imply that your geometry is in some trouble.
Damage implies a non-linear material model of some kind. What were you assuming? Small strain plasticity with isotropic or kinematic hardening? Or a more advanced model like Walker or Chaboche?
Do temperature effects matter to you? Must you do a heat transfer analysis as well?
Do you have a model for metallurgical effects (e.g. austenite/martensite phase changes for carbon steel)? Do you have any heat treatment or grain size data that impact your material model?
I'd recommend starting simple and modeling contact between two teeth, one stationary and another in motion.
I haven't done finite element analysis for a living in many years, but when I was a practitioner this kind of problem would be solved with something like MARC or ABAQUS. I believe ANSYS is very popular now. There are also open source finite element solvers, but I'm less familiar with those.
I'm sure you've done a Google search for something like "finite element analysis gear tooth". You're far from the first to be interested in a problem like this.
Related
I have been trying to make a biologically accurate 2D spatial model of tissue layers, where different physiological processes happen. This includes mainly chemical reactions, diffusion and fluxes over boundaries.
I am making this model in COMSOL Multiphysics, a finite element software package that solves different physics like reaction-diffusion systems, although for my question this might not be really relevant.
In my geometry, I have really small regions between the cells of the tissue layers. These regions serve as openings where diffusion can take place between the cells (junctions). The quality of the mesh is not great here and if I want to improve the quality (mainly by introducing more elements and such), my simulation time increases drastically. The lesser quality mesh also causes convergence to take longer. I added a picture of the geometry to give an idea. I tried different meshes, all with different qualities of the elements and the number of elements ranging from 16000 to 50000.
My background in FEM is really limited and I wanted to know if I can tackle this problem in such a way that it
doesn't negatively affect the biology (keep the tissue domain sizes/problem etc as biologically accurate as possible),
doesn't increase the simulation time drastically,
give a better mesh quality.
So I really want to know what the best way to go is, since I have already thought of some things.
Can I go with the lesser quality mesh (which is not really bad, but not good either), so that I can keep the small regions for optimum biological accuracy and have a relatively small computation time (and hope I won't run into convergence errors).
But maybe there are possibilities that I am missing, for instance: is it possible to make the small domain bigger and then add some kind of factor to the diffusion rates. In other words, if I want to make the domain twice as large, do I factor the diffusion rate with half? Is that even accurate in chemical/physical laws :S.
Hopefully I made the problem a bit clear and thank you greatly in advance for the help.
Cheers,
Mesh of the tissue model
I know this thread was posted some months back but I am unsure if you found a solution.
In order to find the relationship between accuracy and computational time would be that you run a mesh analysis on your model and see how the mesh size directly affects the results you are expecting to obtain (pore pressure, fluid velocity, strain, etc.) This will allow you to determine the most appropriate mesh strategy for your specific problem.
Also, you might need to keep in mind that the diffusion rate of a material will depend on the pore size and the permeability (by means of Darcy's law) so depending on the assumptions you are making for the implementation of your constitutive law and your problem boundary conditions you might simplify/enlarge some of the smaller domains you have in your model so long they are within your previously made assumptions.
I implemented a differential evolution algorithm for a side project I was doing. Because the crossover step seemed to involve a lot of parameter choices (e.g. crossover probabilities), I decided to skip it and just use mutation. The method seemed to work ok, but I am unsure whether I would get better performance if I introduced crossover.
Main Question: What is the motivation behind introducing crossover to differential evolution? Can you provide a toy example where introducing crossover out-performs pure mutation?
My intuition is that crossover will produce something like the following in 2-dimensions. Say
we have two parent vectors (red). Uniform crossover could produce a new trial vector at one of the blue points.
I am not sure why this kind of exploration would be expected to be beneficial. In fact, it seems like this could make performance worse if high-fitness solutions follow some linear trend. In the figure below, lets say the red points are the current population, and the optimal solution is towards the lower right corner. The population is traveling down a valley such that the upper right and lower left corners produce bad solutions. The upper left corner produces "okay" but suboptimal solutions. Notice how uniform crossover produces trials (in blue) that are orthogonal to the direction of improvement. I've used a cross-over probability of 1 and neglected mutation to illustrate my point (see code). I imagine this situation could arise quite frequently in optimization problems, but could be misunderstanding something.
Note: In the above example, I am implicitly assuming that the population was randomly initialized (uniformly) across this space, and has begun to converge to the correct solution down the central valley (top left to bottom right).
This toy example is convex, and thus differential evolution wouldn't even be the appropriate technique. However, if this motif was embedded in a multi-modal fitness landscape, it seems like crossover might be detrimental. While crossover does support exploration, which could be beneficial, I am not sure why one would choose to explore in this particular direction.
R code for the example above:
N = 50
x1 <- rnorm(N,mean=2,sd=0.5)
x2 <- -x1+4+rnorm(N,mean=0,sd=0.1)
plot(x1,x2,pch=21,col='red',bg='red',ylim=c(0,4),xlim=c(0,4))
x1_cx = list(rep(0, 50))
x2_cx = list(rep(0, 50))
for (i in 0:N) {
x1_cx[i] <- x1[i]
x2_cx[i] <- x2[sample(1:N,1)]
}
points(x1_cx,x2_cx,pch=4,col='blue',lwd=4)
Follow-up Question: If crossover is beneficial in certain situations, is there a sensible approach to a) determining if your specific problem would benefit from crossover, and b) how to tune the crossover parameters to optimize the algorithm?
A related stackoverflow question (I am looking for something more specific, with a toy example for instance): what is the importance of crossing over in Differential Evolution Algorithm?
A similar question, but not specific to differential evolution: Efficiency of crossover in genetic algorithms
I am not particularly familiar with the specifics of the DE algorithm but in general the point of crossover is that if you have two very different individuals with high fitness it will produce an offspring that is intermediate between them without being particularly similar to either. Mutation only explores the local neighbourhood of each individual without taking the rest of the population into account. If you think of genomes as points in some high dimensional vector space, then a mutation is shift in a random direction. Therefore mutation needs to take small steps since if your are starting from a significantly better than random position, a long step in a random direction is almost certain to make things worse because it is essentially just introducing entropy into an evolved genome. You can think of a cross over as a step from one parent towards the other. Since the other parent is also better than random, it is more promising to take a longer step in that direction. This allows for faster exploration of the promising parts of the fitness landscape.
In real biological organisms the genome is often organized in such a way that genes that depend on each other are close together on the same chromosome. This means that crossover is unlikely to break synergetic gene combinations. Real evolution actually moves genes around to achieve this (though this is much slower than the evolution of individual genes) and sometimes the higher order structure of the genome (the 3 dimensional shape of the DNA) evolves to prevent cross-overs in particularly sensitive areas. These mechanisms are rarely modeled in evolutionary algorithms, but you will get more out of crossovers if you order your genome in a way that puts genes that are likely to interact close to each other.
No. Crossover is not useful. There I said it. :P
I've never found a need for crossover. People seem to think it does some kind of magic. But it doesn't (and can't) do anything more useful than simple mutation. Large mutations can be used to explore the entire problem space and small mutations can be used to exploit niches.
And all the explanations I've read are (to put it mildly) unsatisfactory. Crossover only complicates your algorithms. Drop it asap. Your life will be simpler. .... IMHO.
As Daniel says, cross over is a way to take larger steps across the problem landscape, allowing you to escape local maxima that a single mutation would be unable to do so.
Whether it is appropriate or not will depend on the complexity of the problem space, how the genotype -> phenotype expression works (will related genes be close together), etc.
More formally this is the concept of 'Connectivity' in Local Search algorithms, providing strong enough operators that the local search neighbourhood is sufficentally large to escape local minima.
Say I am tracking 2 objects moving in space & time,
I know their x,y co-ords and score (score being a probabilistic
measure of the tracked being the actual object),
and I get several
such {x,y,score} samples over time for each object
What metric would I use to measure "similarity" of say a ball moving across a room vs. man moving across the room vs. a child moving across the room.
Assume the score is pretty accurate.
Given your description, I'd recommend looking into a Hidden Markov Model or possibly an artificial neural network or some other machine learning approach. However, there are a number of other techniques that might be more appropriate for your situation.
I'm trying to design a nonlinear fitness function where I maximize variable A and minimize the variable B. The issue is that maximizing A is much more important at single digit values, almost logarithmic. B needs to be minimized and in contrast to A, it becomes less important when small (less than one) and more important when it's larger (>1), so exponential decay.
The main goal is to optimize A, so I guess an analog is A=profits, B=costs
Should I aim to keep everything positive so that the I can use a roulette wheel selection, or would it be better to use a rank/torunament kind of system? The purpose of my algorithm is shape optimization.
Thanks
When considering a multi-objective problem the goal is usually to identify all solutions that lie on the Pareto curve - the Pareto optimal set. Have a look here for a 2-dimensional visual example. When the algorithm completes you want a set of solutions that are not dominated by any other solution. You therefore need to define a pareto ranking mechanism to take into account both objectives - for a more in depth explanation, as well as links to even more reading, go here
With this in mind, in order to effectively explore all solutions along the pareto front you do not want an implementation that encourages premature convergence, otherwise your algorithm will only explore the search space in one specific area of the Pareto curve. I would implement a selection operator that keeps all members of each iteration's optimal set of solutions, that is all solutions which are not dominated by another + plus a parameter controlled percentage of other solutions. This way you encourage exploration all along the Pareto curve.
You also need to ensure your mutation and crossover operators are tuned correctly too. With any novel application of Evolutionary Algorithms, part of the problem is trying to identify an optimal parameter set for the problem domain... this is where it gets really interesting!!
The description is very vague, but assuming that you actually have an idea of what the function should look like and you're just wondering whether you need to modify it so that proportional selection can be used easily, then no. Regardless of fitness function, you should probably default to using something like tournament selection. Controlling selection pressure is one of the most important things you have to do in order to get consistently good results, and roulette wheel selection doesn't allow you that control. You typically get enormous pressure very early, which drives premature convergence. That might be preferable in a few cases, but it's not where I'd start my investigations.
I'm not looking for a general discussion on if math is important or not for programming.
Instead I'm looking for real world scenarios where you have actually used some branch of math to solve some particular problem during your career as a software developer.
In particular, I'm looking for concrete examples.
I frequently find myself using De Morgan's theorem when as well as general Boolean algebra when trying to simplify conditionals
I've also occasionally written out truth tables to verify changes, as in the example below (found during a recent code review)
(showAll and s.ShowToUser are both of type bool.)
// Before
(showAll ? (s.ShowToUser || s.ShowToUser == false) : s.ShowToUser)
// After!
showAll || s.ShowToUser
I also used some basic right-angle trigonometry a few years ago when working on some simple graphics - I had to rotate and centre a text string along a line that could be at any angle.
Not revolutionary...but certainly maths.
Linear algebra for 3D rendering and also for financial tools.
Regression analysis for the same financial tools, like correlations between financial instruments and indices, and such.
Statistics, I had to write several methods to get statistical values, like the F Probability Distribution, the Pearson product moment coeficient, and some Linear Algebra correlations, interpolations and extrapolations for implementing the Arbitrage pricing theory for asset pricing and stocks.
Discrete math for everything, linear algebra for 3D, analysis for physics especially for calculating mass properties.
[Linear algebra for everything]
Projective geometry for camera calibration
Identification of time series / statistical filtering for sound & image processing
(I guess) basic mechanics and hence calculus for game programming
Computing sizes of caches to optimize performance. Not as simple as it sounds when this is your critical path, and you have to go back and work out the times saved by using the cache relative to its size.
I'm in medical imaging, and I use mostly linear algebra and basic geometry for anything related to 3D display, anatomical measurements, etc...
I also use numerical analysis for handling real-world noisy data, and a good deal of statistics to prove algorithms, design support tools for clinical trials, etc...
Games with trigonometry and AI with graph theory in my case.
Graph theory to create a weighted graph to represent all possible paths between two points and then find the shortest or most efficient path.
Also statistics for plotting graphs and risk calculations. I used both Normal distribution and cumulative normal distribution calculations. Pretty commonly used functions in Excel I would guess but I actully had to write them myself since there is no built-in support in the .NET libraries. Sadly the built in Math support in .NET seem pretty basic.
I've used trigonometry the most and also a small amount a calculus, working on overlays for GIS (mapping) software, comparing objects in 3D space, and converting between coordinate systems.
A general mathematical understanding is very useful if you're using 3rd party libraries to do calculations for you, as you ofter need to appreciate their limitations.
i often use math and programming together, but the goal of my work IS the math so use software to achive that.
as for the math i use; mostly Calculus (FFT's analysing continuous and discrete signals) with a slash of linar algebra (CORDIC) to do trig on a MCU with no floating point chip.
I used a analytic geometry for simple 3d engine in opengl in hobby project on high school.
Some geometry computation i had used for dynamic printing reports, where was another 90° angle layout than.
A year ago I used some derivatives and integrals for store analysis (product item movement in store).
Bot all the computation can be found on internet or high-school book.
Statistics mean, standard-deviation, for our analysts.
Linear algebra - particularly gauss-jordan elimination and
Calculus - derivatives in the form of difference tables for generating polynomials from a table of (x, f(x))
Linear algebra and complex analysis in electronic engineering.
Statistics in analysing data and translating it into other units (different project).
I used probability and log odds (log of the ratio of two probabilities) to classify incoming emails into multiple categories. Most of the heavy lifting was done by my colleague Fidelis Assis.
Real world scenarios: better rostering of staff, more efficient scheduling of flights, shortest paths in road networks, optimal facility/resource locations.
Branch of maths: Operations Research. Vague definition: construct a mathematical model of a (normally complex) real world business problem, and then use mathematical tools (e.g. optimisation, statistics/probability, queuing theory, graph theory) to interrogate this model to aid in the making of effective decisions (e.g. minimise cost, maximise efficency, predict outcomes etc).
Statistics for scientific data analyses such as:
calculation of distributions, z-standardisation
Fishers Z
Reliability (Alpha, Kappa, Cohen)
Discriminance analyses
scale aggregation, poling, etc.
In actual software development I've only really used quite trivial linear algebra, geometry and trigonometry. Certainly nothing more advanced than the first college course in each subject.
I have however written lots of programs to solve really quite hard math problems, using some very advanced math. But I wouldn't call any of that software development since I wasn't actually developing software. By that I mean that the end result wasn't the program itself, it was an answer. Basically someone would ask me what is essentially a math question and I'd write a program that answered that question. Sure I’d keep the code around for when I get asked the question again, and sometimes I’d send the code to someone so that they could answer the question themselves, but that still doesn’t count as software development in my mind. Occasionally someone would take that code and re-implement it in an application, but then they're the ones doing the software development and I'm the one doing the math.
(Hopefully this new job I’ve started will actually let me to both, so we’ll see how that works out)