iMechanica - materials modeling - Comments

Ezio Bruno Deparment of

Ezio Bruno — Tue, 18 Nov 2008 13:56:26 -0500

Ezio Bruno Deparment of Physics, University of Messina

Dear Vikram,

thank you very much for this interesting blog and congratulations for your very nice research work. I am really interested in. I'm also trying to go to million atoms following a very different path, though again the starting point is Density Functional theory. If you are interested in, here are the references to some papers of my group on the subject:

E. Bruno, F. Mammano, N. Fiorino and E.V. Morabito, "Coarse-grained density functional theories for metallic alloys: Generalized coherent-potential approximations and charge-excess functional theory", Phys. Rev. B 77, 155108 (2008) doi:10.1103/PhysRevB.77.155108

E. Bruno, F. Mammano and B. Ginatempo, "Coarse grained Density Functional theory of order-disorder phase transitions in metallic alloys", arXiv:0810.5367v1 [cond-mat.mtrl-sci]

How is the work on extending this approach to K-S DFT going on

Teng zhang — Fri, 10 Oct 2008 21:52:07 -0400

Dear Vikram

Very interesting work. I want to know How the work on extending this approach to K-S DFT get along with and what the difficulties in the extending. To be honest, I am not familiar with the theory DFT, so my questions may be so simple.

I have read some literatures about the defect state of photonic crystals, unfortunately, we did not get much progress and gave up for some reasons. However, I am still interested in this area, so I want to konw the new ideas (especially those based on the computational/ soild mechanics).

I hope to hear your replies and thanks in advance .

How to model Aluminium Metal Matrix Composites using ABAQUS CAE

ksreddy — Wed, 18 Apr 2007 00:38:12 -0400

Dear Sir,

Iam a ressearch scholor from IITM,I would like to know procedure that,how one can model particulate metal matrix composites (wherein reinforces particles of different sizes,orients in different directions in the metal matrix) using either ABAQUS CAE or ANSYS.If you have any paper on the releavant matter ,please send it

I will be very greatful,if you provide this information.

Re: Further questions on DFT

Vikram Gavini — Mon, 05 Mar 2007 09:14:09 -0500

Dear Ajit,

1. The reasons we didnt use FD was because

(i) FD doesnot respect the variational structure of the problem

(ii) As you mentioned it cant handle arbitrary goemetries and boundaries

(iii) And, more importantly, it cant handle unstructured coarse-graining, which is the heart of our multi-scale scheme. Such a coarse-graining allows us to handel multi-million systems using OFDFT, which has not been possible to date. Such a scheme is important in simulating systems where the physics on the quantum scale as well as macroscopic scale are important, for ex. defects in materials as well as some applications you mentioned. Please refer to our second paper for more information.

Every variational formulation has an equivalent strong formulation (differential formulation), ofcourse assuming certain regularity in the system (but vice-versa is not true). The PDE associated with a variational formulation is the Euler-Lagrange equations. So DFT, in particular, has a differential formulation too.

I am familiar with one work recently (there have been some in the past too), where FD scheme is used to discretize DFT. Please refer to, "An Efficient Real Space Method for Orbital-Free Density-Functional Theory", C.J. García-Cervera, Comm. Comp. Phys., 2 (2), pp. 334-357 (2007).

Regarding Gallerkin formulation, as I mentioned before, a discrete variational formulation and Gallerkin formulation are equivalent (just like a variational formulation, weak-form are equivalent).

2. Yes, the three are related. We can discuss more on personal correspondence if this is not clear. Yes, DFT can handle glass at 0K.

3. Motion can be handled in DFT, though periodicity imposes serious restricions in modelling such systems. The physics of DFT is very fundamental, its the discretization and numerical schemes which pose the challenge.

4. DFT is derived from QM for ground-state properties and has all the features of QM. So, I am speculating it will be able to model the bucky-ball through a small slit though I havent come across this.

Further on the DFT-FEM questions

Ajit R. Jadhav — Mon, 05 Mar 2007 01:41:13 -0500

Hi Vikram,

Yes, the answers were helpful, but sometimes only in part.

1. I raised the question about FDM because: (i) I love its simplicity!; and (ii) the most important advantage of FEM--its ability to better handle more complicated BCs within a uniform framework--seems to have been absent here anyways (though I can't immediately see why).

Is it the case that DFT has not been formulated in terms of an equivalent differential equation at all? Is this the basic reason why FDM and the specifically Galerkin form of FE formulations have not been pursued thus far?

Not having a differential formulation of DFT is unexpected considering that (i) it is the variational form which is more restrictive, (ii) DFT work received a Nobel, and (iii) the way physics is usually done, whereby even a minor nuance of an alternative approache is not left unpursued. Possible, but unexpected.

2. A dumb question, again! What does periodicity basically mean in this context? The lattice-like regular arrangement (periodic in real space)? The particular form of BCs (their symmetry)? The basis in plane-waves (sort of like Fourier expansion)? Do the three have to be related necessarily?... It wouldn't seem to be so on the face of it... Some of my questions also probe these aspects... For instance, let me ask you one more question: You mention non-zero temp. OK. So, can DFT-FEM handle glass at 0 K (as a theoretical scenario)? How about a solid solution with randomly placed second component?

I did note Pradeep Sharma's comment about the freedom from periodicity that your work now allows. So your work does seem to be important. Yet, as an outsider, I can't quite see why the plane-wave basis would be a limitation... Just thinking aloud--you may leave this point alone.

3. Then, my questions also dealt with the aspect of *motion* within the system: whether and how DFT (and its FEM approximation) could handle motion. Yours seems to be a static model. Or is it the case that the inability to handle non-periodic BCs has also meant, so far, the inability to model motion of atoms/ionic cores?

4. Finally, my question on bucky-ball through a narrow slit also probed the power and suitability of the DFT approach to address (i) tunneling and (ii) wave-particle duality of matter. These are the routine problems for QFT folks to ponder on. How does DFT fare here? How well would a DFT-FEM simulation work out here?

In the present post, the question numbers 1 and 4 are more immediately interesting to me than the others.

Re: General questions on DFT

Vikram Gavini — Sun, 04 Mar 2007 11:45:13 -0500

Dear Ajit,

I am answering your questions in brief below.

1. DFT itself is a variational formulation. It states that "The ground state properties of a system depend only on the electron-density". This is a variational statement. Now, FEM is better than other approaches like FD because it respects the variational structure of the problem. A discretized version of the variational formulation and a corresponding Gallerkin formulation both result in the same set of equations.

2,3,4 : Some of these applications are possible and are being attempted. However, there are two main obstacles which appear in most applications you mentioned.

(i) In most such applications one needs to break away from periodicity, which has been a serious restriction till date as most DFT codes are written using a plane-wave basis. This is where a real-space formulation is useful which can handel more general non-periodic systems.

(ii) Some applications you mention require simulations at non-zero temperature (higher temp) and this is still a holy grail problem in electronic structure calculations.

I hope these answers are useful.

General questions on DFT itself

Ajit R. Jadhav — Sun, 04 Mar 2007 00:39:10 -0500

Hi Vikram and other mechanicians,

I am posting some questions of general nature on DFT here. Yet, I want to note right at the outset that DFT is not my area of special knowledge let alone of research. My interest in DFT (as of today) mostly is in the nature of a layman's curiosity. May be that's why I have a lot of naive (perhaps even dumb and stupid!) questions to ask about DFT.

I am posting these questions here, as a comment to Vikram's post, even though these aren't exactly comments on his two forthcoming papers. I am sure those papers is what he would like to have the discussion here mainly focused on. Yet, these questions of mine do pertain to DFT. So, I decided to post them here... If these questions become interesting enough, anyone may please feel free to create separate discussion threads and post them there... I hope Vikram won't mind thus starting out this discussion on the DFT-related points off his post.

(1.0) Why do people use the variational formulation of FEM for DFT? Two sub-questions...
(1.1) Why do they not use simple FDM (finite difference method)?
(1.2) Why not use Galerkin's method?

(2.0) The difference between DFT and ab initio calculations seems to be well noted in the literature. But what is the main difference between QFT and DFT?
(2.1) Why is DFT not used to analyze superconductivity?

All the following questions have this prefix: "Using the currently available DFT simulation technology..."

(3.0) Can one simulate ordinary liquids, say, water or molten tin or copper? Is there any salient difference between using DFT for solids vs. for liquids?
(3.1) Can one predict thermodynamic quantities like, say, heat of fusion?
(3.2) Can one model nucleation and growth in solidification?

(4.0) Can one model the motion of, say, a small cluster of water molecules against a backdrop of other millions of water molecules?
(4.1) Will it be possible to simulate some liquid passing through a nanotube? Can one think of any major obstacle against modeling this situation--something that would be very obvious to a DFT expert?
(4.2) Will it be possible to simulate a buckyball passing through a narrow slit? Will this simulate diffraction--as has been experimentally observed?

Thanks in advance for all your replies.

Re: FEM and DFT

Vikram Gavini — Sat, 03 Mar 2007 21:31:30 -0500

Dear Sukumar,

Thankyou very much for your comments. I am looking forward to reading your papers.

FEM and DFT

N. Sukumar — Sat, 03 Mar 2007 20:22:22 -0500

Read with interest your papers. Very nice and comprehensive-work that brings together tools and techniques from different fields along with high-performance computing in an attempt to `bridge the scales.' The essentials on OF-DFT are well-explained; am more familiar with KS-DFT using the pseudopotential approximation, and also some elements of Gamma/epi-convergence. Per what I have seen in the literature, most multiscale methods typically use empirical potentials and few start from first principles as you have proposed. I saw Pask's work on FE-DFT cited, and he is one among just a handful who are involved in the development and application of finite elements in self-consistent KS-DFT calculations. We're currently applying partition-of-unity finite elements for the Schrodinger equation (eigenproblem), and results are very encouraging (vis-a-vis other real-space methods and even planewaves). Will post/discuss the same when the papers are realized in the coming months. To create a link, you can highlight the URL (go into edit) and then click on the `chain' symbol in the menu and provide the URL there too.

Re: quite interesting

Vikram Gavini — Fri, 02 Mar 2007 10:39:51 -0500

Dear Pradeep,

Thanks! In response to your questions -

(1) We have developed the QC approach for orbital-free density-functional theory (OFDFT). OFDFT is a reliable computational tool for systems with electronic structure close to that of a free electron gas (like Al, simple metals ..), but can give erroneous results for systems which are covalently bonded. Hence, at this point we can't study semi-conductor devices. The most reliable computational tool for any material is the Kohn-Sham version of density-functional theory (KS-DFT). I am working on extending the present approach to achieve coarse-graining for the KS-DFT too. Once this is done, then we will have access to all materials and all properties (structural, optical, electronic, magentic ...)

(2) The standard quasi-continuum codes developed so far only handle the coarse-graining of the displacement field. But in DFT we need to coarse grain the electronic fields too, which is non-trivial. In our second manuscript, titled "Quasi-continuum orbital-free density-functional theory: A route to multi-million atom DFT calculation" (which is available as article under press on the JMPS website as well as a link is posted on my blog) we show how we achieve this. In short, we use a predictor-corrector approach and construct a numerical scheme which is supported by a theorem by Lions et. al (From Molecular models to continuum mechanics, ARMA 164 341, 2002). The non-periodic real-space approach and the finite-element discretization of OFDFT from our first manuscript ("Non-periodic finite-element formulation of OFDFT") are key to developing this scheme.

I will be really happy to discuss more with you.

quite interesting

Pradeep Sharma — Fri, 02 Mar 2007 08:20:37 -0500

Vikram,

This is quite interesting. I did notice one of your papers that appears in the "Articles in Press" section of JMPS. The freedom from periodic boundary conditions is nice.

I have two simple questions: (1) Have you tried using/testing your methodology to study semiconductor clusters (quantum dots) from the viewpoint of electronic structure? (2) Did you implement your work within the quasicontinuum code available online?

Details and Applications

Vikram Gavini — Thu, 01 Mar 2007 23:07:25 -0500

Dear Henry Tan,

This work has not yet appeared in print, but is available online on the JMPS website as articles in press. For some reason my links in the post didn't appear. I have updated my post now. I am also giving the links to these papers here.

http://www.its.caltech.edu/~vikramg/academic/OFDFT-FE.pdf

http://www.its.caltech.edu/~vikramg/academic/DFT-FE-QC.jmps.pdf

Regarding applications of this approach, there are many. This methodological development seamlessly couples the quantum mechanical and continuum length scales with density-functional theory as its sole physics, which is the most reliable, computationally feasible and transferable theory to compute materials properties. So, broadly speaking, the applications are in studying materials with defects from a quantum mechanical perspective. Right now, I am looking at two applications: (1) Studying the problem of vacancy clustering and formation of prismatic loops in aluminum, which is important from the viewpoint of radiation damage in materials. We have observed some very interesting physics over here which have not been observed before (this work is not published, so I am unable to give you details; but will keep you posted) (2) Studying a single dislocation resolving all length-scales and computing the core-energy.

The other examples which I can list are : computing migration energies of these defects, studying surface reconstructions, phase transformations in materials. Once we extend this approach to the Kohn-Sham version of density-functional theory, one can study how optical, electronic and magnetic properties of complex materials (like ferro-electric, photonics materials) get influenced by defects like surfaces, domain walls and others.

There have been attempts to study these problems before, which broadly fall under the category of multi-scale modeling. But all these approaches have used heterogeneous theories to describe features at different length scales and various hypothesis are used to stitch these heterogeneous theories across the boundaries. This work differs from those approaches in an important way. We use the same physics everywhere in the domain (OFDFT) and use the power of coarse-graining to seamlessly transition from quantum mechanical length scale to continuum length scales. So we keep the same fundamental physics everywhere in the domain and play with the numerics.

I will be really happy to discuss more if you are interested.

applications and details

Henry Tan — Thu, 01 Mar 2007 20:47:41 -0500

Can you give an example: in what applications will this multiscale quasi-continuum orbital-free density-function calculation be applied to. Thanks.

Is the work published? I am very interested in reading some details.

Please add a hyperlink to the publisher's webiste

Zhigang Suo — Sat, 16 Sep 2006 13:35:55 -0400

Could you please add a hyperlink to the publisher's webiste? Or perhaps link to the page of the book on Aamzon. The book may attract reader reviews at some point. That information is helpful to other readers. In any case, Amason should have more information about the book.