Why is it important to have a Weak Formulation for FEM and why it does not give accurate results? What type of method and techniques are available to get accurate results using weak formulation?
Solving the strong form (governing differential equations) is not always efficient and there may not be smooth (classical) solutions to a particular problem. This is true especially in the case of complex domains and/or different material interfaces etc. Moreover, incorporating boundary conditions is always a daunting task with solving strong forms directly. The requirement on continuity of field variables is much stronger.
In order to overcome the above difficulties, weak formulations are preferred. They reduce the continuity requirements on the approximation (or basis functions) functions thereby allowing the use of easy-to-construct and implement polynomials (for example widely used Lagrange polynomials). This is one of the main reasons for the popularity of weak formulations despite many disadvantages they pose when applied some class of problems, like non-self-adjoint systems (non-symmetric matrix systems) and advection dominated fluid flow (require stabilisation techniques to get accurate solutions). In weak forms, Neumann boundary conditions come naturally and hence, implementing them is very easy (though they are satisfied in a weak sense.)
Weak forms never give (perfectly) accurate solutions because of the reduction in the requirements of smoothness and weak imposition of Neumann boundary conditions. But this comparison is valid only when you compare the weak solutions with the classical solutions. Weak forms still give relatively very accurate results with the mesh refinement, which are extremely good for engineering simulations; and you will get a solution even if there is no 'classical' solution (in case of problems with complex domains and different materials, contact etc.)
Improving the accuracy of a solution in weak formulations depend upon the type of problem you are solving. In some cases, for example, elliptic problems, only mesh refinement is good enough and but when weak formulations are applied to advection-diffusion, Stokes and Navier-Stokes flows, one needs to use efficient stabilisation techniques, along with mesh refinement, to get accurate results. The accuracy can also be improved by using higher-order shape functions.
Guess you talk of second order boundary value problems, and putting it in short, let's take the Possoin equation as an example: One wants to solve simultaneously two (partial) differential equations and the constitutive law. In this case using the classical languag one wants to solve, div F = q, curl G = 0, and F = alpha G. In the modern language one would write dF = 0, dG=0, and F = *alpha G, where * is the so called Hodge operator.
A finite element solution is about seeking for a solution for these equations in finite dimensional spaces spanned by the chosen finite element basis functions, and this is what creates the issue: These equations do not have together a solution in the typically chosen finite dimensional spaces. Hence, instead one seeks for an approximative solution. To find such a solution, one expresses the other diff. eq. in the weak form relaxing the system enough to have a solution.
FEM are based on the weak formulation because this relation represents, typically but not always, a variational principle satisfied by the solution. Think for example of the Poisson equation in a bouded domain, completed with homogeneous Dirichlet boundary conditions (b.c.): one can either address the strong form directly (- Laplacian u = f, + b.c), as is done, e.g., with the finite difference method, or one can deal with the weak form, which actually represents the principle of minimization of the total energy: integral over the domain of (0.5 |grad u|^2 - f u). This minimization yields the principle of virtual works: seek u in U (a suitable function space) such that: integral over domain of (grad u)* (grad v) - f v = 0 for all test functions v in U.
It turns out that the regularity required for u in this last case is weaker than the one associated with the strong form, e.g., very crudely, you need second order derivatives in the strong formulation and only first order derivatives in the weak formulation.
FEM are based on this very same formulation, except that the both the solution and the test functions belong to a suitable finite dimensional subspace of U.
Of course I am skipping many theoretical and practical issues which are really involved to tackle here, but that one should be aware of.
I don't see the point of inaccuracy that you mention: FEM yields an approximate solution whose accuracy depends on the regularity of u and on the properties of the finite elements at hand, e.g, polynomial degree and mesh spacing. For example, for a very smooth solution u, if you can afford to let either the degree grow or the mesh spacing shrink, you can reach whatever accuracy you want to.
On the other hand, for some problems, such as advection-diffusion equations with a strong advective field, it is well known that the accuracy of the solution may be very poor if the mesh size is not sufficiently small. But remedies to this state of affairs are also very well known.
Just to complete the above answers, the weak formulation relaxes the continuity requirements for the approximate displacements since you pass derivatives from the the approximate solution to the weighting function. This do not have any inaccuracy issue aside from the approximate nature of the solution itself.
There is no contradiction. When one solves the Poisson equation in some bounded domain with finite elements one seeks for a solution from some *finite dimensional* space. As soon as the boundary value problem is "finite dimensionalized", there is a cost (except in some special cases). The solution of the original Poisson problem is in the space of square integrable functions whose gradients/exterior derivatives are also square integrable. Although the FE-solution convergences towards the correct solution –and hence, in principle any accuracy is possible– it's still not the same solution, as one is not able to express all the functions of the original space with linear combinations of a finite amount of basis functions.
In case of finite differences –and in case of all Yee-kind of techniques in wave propagation– one imposes both differential equations in the "strong sense". But now, the issue is shifted to the constitutive law, which is not imposed on all regular points of the domain. The cost of this is, FDTD and Yee-like techniques are restricted to meshes having some sort of orthogonality between the primal and dual grids. That is, the other equation –such as curl E = 0 in electrostatics– is imposed on the primal and the other, –here div D = q, resp.– on its dual mesh. A constitutive law kind of relation –here D = eps E, resp.–is exploited to impose a relation between the two differential equations on the finite amount of nodes where the primal and dual mesh intersect. (Notice, curl E = 0, div D = q, D = eps E amounts in saying div eps grad Phi = -q.)
Weak form means, instead of solving a differential equation of the underlying problem, an integral function is solved. The integral function implicitly contains the differential equations, however it's a lot easier to solve an integral function than to solve a differential function. Also, the differential equation of system poses conditions that must be satisfied by the solution (hence called STRONG form), whereas, the integral equation states that those conditions need to be satisfied in an average sense (hence WEAK form). However, do not understimate the power of integral functions just by its name "weak form".
An example is stresses on free surface. Strong form says if there is no traction on the surface, corresponding stresses must be zero.So if we can solve the PDE, we'll get exact zero stresses. But all FEA code will give you some stress on the surface since whole FEA is based on integral functions. So as you refine the mesh, the stress value at traction free surface goes closer and closer to zero...this is average sense
A weak form is just a alternative way and more convenient of stating the underlying mathematical problem and is more general than the differential equation form. (They may be equivalent if some requirement of regularity of the solution is satisfied.) The weak form also allows for approximate solutions with weaker requirement on continuity, e.g. the derivatives may not have to be continuous. The finite element method is a conceptual way of constructing a finite set of global basis functions in a function series. (by putting together element local functions). This function series ansatz is plugged into the weak form and the system matrices come out on the other side. What is characterizing of these basis functions is that they are non-zero over only a part of the domain, namely over a patch of adjacent finite elements. The choice of basis functions could be more or less efficient with respect to the problem solved.
I would answer with another question; How can you demonstrate that the approximated solution you get by FEM converge to the mathematically exact solution?
If you want to answer to my question you end up with a weak formulation that allow you to get a broader range of solutions that are closer to the physical real world .
Solving the strong form (governing differential equations) is not always efficient and there may not be smooth (classical) solutions to a particular problem. This is true especially in the case of complex domains and/or different material interfaces etc. Moreover, incorporating boundary conditions is always a daunting task with solving strong forms directly. The requirement on continuity of field variables is much stronger.
In order to overcome the above difficulties, weak formulations are preferred. They reduce the continuity requirements on the approximation (or basis functions) functions thereby allowing the use of easy-to-construct and implement polynomials (for example widely used Lagrange polynomials). This is one of the main reasons for the popularity of weak formulations despite many disadvantages they pose when applied some class of problems, like non-self-adjoint systems (non-symmetric matrix systems) and advection dominated fluid flow (require stabilisation techniques to get accurate solutions). In weak forms, Neumann boundary conditions come naturally and hence, implementing them is very easy (though they are satisfied in a weak sense.)
Weak forms never give (perfectly) accurate solutions because of the reduction in the requirements of smoothness and weak imposition of Neumann boundary conditions. But this comparison is valid only when you compare the weak solutions with the classical solutions. Weak forms still give relatively very accurate results with the mesh refinement, which are extremely good for engineering simulations; and you will get a solution even if there is no 'classical' solution (in case of problems with complex domains and different materials, contact etc.)
Improving the accuracy of a solution in weak formulations depend upon the type of problem you are solving. In some cases, for example, elliptic problems, only mesh refinement is good enough and but when weak formulations are applied to advection-diffusion, Stokes and Navier-Stokes flows, one needs to use efficient stabilisation techniques, along with mesh refinement, to get accurate results. The accuracy can also be improved by using higher-order shape functions.
@Roberto Bernetti :
That's a very good question.
One can make use of functional analysis to prove the converge of error norms for the given bilinear and linear forms or calculate error norms numerically and verify the convergence. But this is not particular to weak formulations. This can be applied to any other finite element formulations.
Oh, for real-life applications involving PDEs in 3D, the FEM is the only possible solution method. Usually, an accuracy of 10^{-3} is enough for engineering applications, which is easily achievable. As for other methods using the weak formulation, there are many but integral equation methods and boundary integral methods are important examples in this class.
Anyway, ALL finite element formulations came from the weak formulation.
@Eric:
Not ALL finite element formulations are from the weak formulation. There are other class of formulations as well. Rayleigh-Ritz method and Petrov-Galerkin methods are there. And least-squares based finite element formulations are getting popular now-a-days because of their advantages over weak formulations.
No. Like I mentioned in my previous comment. Not the LEAST-SQUARES based formulations.
Dear abdullah
Fem is developed form of Galerkin, so if you use weak form, you have an integral equation which have lower degree of integral and as a result the calculation get easier.In Galerkin method the shape function equal to weight function so in this condition the main aim is to have one degree of differential ( the degree of weight equal to shape function ) with this trick you can use lower degree of shape function. It is noteworthy that with weak form the degree of equation goes down and the degree of weight function goes up.
In today's practice of FEM, we have forgotten the convergence nature of the weak form so we go to solutions of 200 million dof. My confidence in the weak form is enhanced by its ability to reflect the various instabilities in the equations such as limit load, fracture mechanics, vibrations when compared to experiment.
Weak formulations enable better finite element approximations. It is an approximate solution which is close to or almost approximate to the exact solution. Method of weighted residuals( Galerkin method, collocation method, moment method, sub-domain method,least square method) and Rayleigh-Ritz method can be employed.
First of all, FEM is defined as a discretization of a weak form of some PDE. The main advantage is, that in the weak form we do not need 2nd classical derivatives for a 2nd order PDE (the kind appears at most in practice) to formulate a solution, because the weak form can be solved by a function coming from a Sobolev space H^{1,2} or so, meaning, we only need 1st weak derivative. This simple makes the space where the possible solution could "live" way "larger" increasing the chance to find such a solution.
One could try to solve something really simple, for instance a one dimensional wave equation without any coefficients in a classical sense and will see, that the solution one gets by some basic analytical approach (usually for this example one makes a coordinate transformation like eta=x-y, xi=x+y and rewrites the equation in the sense of new coordinates) does not satisfy the regularity condition of the equation (in this example we can get a solution which can be derived only once, but not twice). This solution could be a solution in weak sense.
At this point correct me if I am wrong, but I remember, that at least for some PDEs (for instance for heat transfer in linear case) the weak solution converges to the classical solution, if such exists. For PDEs where the classical one does not exist it makes no real sense to discuss whether or not a weak form is good or not good...
As for error estimates and such mentioned earlier, there are not really sth. coming from the weak form itself but from the discretization. Discretization in FE-sense means, that we create a finite dimensional subspace of infinite dimensional Sobolev space (one can't really do numerical approaches on infinite dimensional spaces). The solution we get from the FE can be seen as an projection of the solution in the Sobolev space to our discrete space with less dimensions. Both, a-priori and a-posteriori error estimates work on discrete subspaces, not on Sobolev space itself. One can try to visualize this by a simple example: assume, that you have a function from R^2 to R and you are looking for instance for it's minimum (or maximum, does not matter). Assume that you can not do any calculations in R^2 but you still wanna get some solution, even if it is not exact. Than you can choose a subspace of R^2 with less dimensions (i.e. 1...), which would be a line (not necessary an axis). Along this line you can then calculate the minimum of the function, which highly depends on where you draw the line, if you are lucky you might even get the exact solution if you hit the minimum in R^2, but usually you won't. Each line you try is it's own subspace, in FE-sense we would speak of them as different discretizations (the error in the solution depends on the discretization you choose).
I just want to add that meshfree methods are also based on weak formulations. So, the conclusion is that numerical methods should be based on weak forms. Note that collocation methods, which would be more efficient than weak form based methods, are not stable and thus not useful.
when we have a Weak Formulation for FE. It is lest work than a strong formulation. The space of weak solution is more big that the space of strong solution. The strong solution this more near to the exact solution.
It is worth mentioning that the finite volume method (FVM) operates directly on the strong form of the governing equations. Although FVM is synonymous with computational fluid dynamics, it is gaining popularity in other fields such as computational solid mechanics and can be applied efficiently and accurately to linear/nonlinear mechanics.
Q:Why is it important to have a Weak Formulation for FEM
A: It is necessary, in order to treat the essential and natural boundary conditions in a correct manner.
Q:and why it does not give accurate results?
A: It gives the best possible result. Perfect accuracy is a dangerous dream :)
Q: What type of method and techniques are available to get accurate results using weak formulation?
A: Adaptive FEM
Strong form contains second order derivatives, whereas weak form has only first order. In the strong form, the material constants should be smooth enough to be derivable, whereas in the weak form they can be even non-continuous.
In the strong form, boundary conditions are added to the equation, whereas in the weak form it is included in the formulation (essential and natural b.c. are identified by the weak form only).
The PDE is locally imposed, in every point of the computational domain, whereas the weak form is global (checked with the test functions). So the best global approximation may be generated.
Weak formulation is more appropriate to check the problem correctness (solution existence, uniqueness and its continuity w.r.t. input data), by using the modern functional framework. This formulation is quite natural within the Functional analysis.
For the most practical problems, weak formulations generate better and more accurate numerical solving methods than strong formulations.
FOr many problems, FEM solution based on weak formulation is mathematically proven to be the optimal one (with a minimal distance from the exact solution).
To add my cent to the discussion, I see the FEM as:
1. A special case of projection methods. With that in mind I think most of the techniques mentioned can be formulisable in that framework. Another interpretation is simply a generalised average formulation. Once we have the weak form, then choosing the base (projection) functions will lead to special cases as FEM, FVM, co-locations (with Dirac distributions), etc.
2. Going to more physics, the "weak" form is generally the base form for mass conservation, flows, mass transfer etc given a control volume (CV). This makes it more general than PDE.
3. PDE are generally obtained from 2 by taking some limit (CV going to zero), and under some continuity assumption the corresponding strong form is then obtained. Jump conditions [Rankine-Hugoniot] might be required if discontinuities are present. This is handled when the limit is taken to ensure that conservation is honored.
4. Note that the fundamental theorem relating both formulations is the divergence theorem (or Gauss's theorem or Ostrogradsky's theorem) which allows going from an integral on the surface [natural boundary condition] to the volume delimited by that surface.
5. Numerically, FEM is generally not suited for flow simulations as it has mass conservation issues if not care is taken. For flow simulations FVM are generally more suited as they intrinsically contain that mass conservation aspect.
6. It looks like there are several lines of thought, one advocating FEM is the way [this is maybe pushed by structure mechanics], one advocating FVM is the way to go [CFD community]. When fluid-structure are coupled that gives an interesting mix. Personally I prefer FVM.
I hope this helps.
Reasons [from the intro of every FEM book]:
1. automatically enforces natural BCs.
2. automatically enforces physically correct jump conditions at internal interfaces [e.g., where coefficients have discontinuities]
3. reduces the order of continuity needed for elements selected.
4. when the solution seeks a minimum energy configuration, preserves this property in the discrete approximation.
as for accuracy, with a good method, high accuracy good mesh. It gives high accuracy with a good mesh and can even construct such a mesh by itself (self adaptively).
FEM is a good tool. like any tool, it is not for every problem / application.
You can find the answer to your question in the following papers.
1st paper
Article Soft Core Plane State Structures Under Static Loads Using GD...
2nd paper
Article GDQFEM Numerical Simulations of Continuous Media with Cracks...
3rd paper
Article On Static Analysis of Composite Plane State Structures via G...
4th paper
Article Generalized Differential Quadrature Finite Element Method fo...
In compare to strong formulation, in weak formulation (i.e. FEM) for solving the problem requires lower order continuity in the basis function space.
Strictly saying any physical problem can be formulated
either in form of PDE with corresponding initial (for dynamics) and boundary conditions = strong form,
or in form of variational equations = weak form.
Both formulations are equivalent in the proper functional spaces.
FEM is just a numerical method to solve the problem - roughly speaking -
the FEM is based on special discretization techniques of the weak formulation and further numerical solution of corresponding linear or non-linear system.
So the weak formulations is the setup of the physical problem, but not a method.
The error in FEM depends on many factrors (e.g. for statics -- order of shape functions used during discretizations, size of the mesh) - for the case you need one can consult special books about error estimation for FEM (earliest and most famous with Ivo Babuska)
Starting with the weak form one can obtain
necessary equations for further numerical methods like Boundary Element Method, Meshless Method etc etc
(e.g. one can come to BEM using the fundamental solution of the strong formulation + further transformation of the variational equations)
In the following another demonstration of the accuracy of strong formulation. The results are documented by the following paper.
Article Generalized Differential Quadrature Finite Element Method fo...
The variational formulation casts the problem in integral (compact form).
For example if we are to solve the Dirichlet problem on a finite domain with homogeneous b.cs, then the variational formulation casts the problem in the appropriate Sobolev space, whose embedding in L2 is compact (Rellich's theorem). Lax-Milgrams theorem assures the estence of a solution in this case.
Saying that FEM does not give accurate results is an incomplete and incorrect question.
If you think about coarse meshes and displacement-based FEM it is obvious that FEM does not give accurate results, because they use low order polynomials for the approximation of the solution.
If you want to use weak formulation you must use other types of FEM, such as SEM (Spectral Element Method) o p-FEM that can use polynomials of high-degree inside each element and give you more accurate results with a coarse mesh.
However, as stated by previous papers that I posted here, I think that strong formulation based element method can give you a very high accuracy for the smallest number of elements in the mesh, due to also the C1 continuity conditions among the elements, a property that you don't generally have in standard FEM where C0 continuity is enforced.
We're working on a paper about the comparison between weak and strong formulation, I'll certainly post our results here when they'll get published.
Really, and what if the problem does not have a classic solution ?
The real answer to your question is to be found in history and might be the following:
when the finite element technique has been developed (about 1950-60) we were able to do very well the numerical integrals and then through the weak formulation we were able to solve the problem. In fact at the time for the calculation of the derivatives at one point there was only the finite difference technique which as everyone knows has an accuracy that is not too high. The other problem was that the boundary conditions must be imposed in the string formulation point by point, while in the weak do not.
With the advent of differential quadrature (1972) and its development (1992 and 2000) the accuracy in the calculation of the derivatives increased significantly and the strong formulation is currently widely used because, as far researchers have already shown, it is more accurate.
Some of this points can be seen in my papers.
Best Regards,
Francesco Tornabene
The fundamental difference between FEM and FD is that the former provides an approximation of the *solution* of a PDE while the latter is based on approximations of the *equation*. Using a weak form is the natural choice for FEM in this sense - the shape functions act as test functions used to optimize the numerical approximation of the solution.
The main advantage of the weak formulation in the FEM is the ability to deal with non smooth solutions of the problem. It can be noted that the local approximation is not supported when using the finite-difference approximations in the cases where solutions do not have a sufficient number of derivatives. Another point is the way to define the inner product (inner product is defined in the same way as the weak solutions) to get the orthogonality of the residual to the vector space of the solutions.
FEM is a special type of Ritz's direct method. Therefore, a variational formulation of the field equations of the respective problem is essential. The so-called weak formulation is a kind of default way to construct a variational formulation.
Since a variational formulation is usually of lower order than the original equations, the solutions of it need not to be as many times differentiable than the solutions of the original equations. Therefore, the weak formulation may have more solutions than the original equations.
In the following another demonstration of the accuracy of strong formulation. The results are documented by the following paper.
Article Strong Formulation Finite Element Method for Arbitrarily Sha...
In the following another demonstration of the accuracy of strong formulation. The results are documented by the following paper.
Article Strong Formulation Finite Element Method for Arbitrarily Sha...
What you call a 'Weak' formulation must be understood as a 'Flexible' formulation of your mathematical model: this latter requires (or demands) to be exactly verified on each point of your domain. You will naturally agree that this requirement is strong enough to be called a Strong form. A weak formulation is a way to relax this statement that will permit to verify it in an averaged way, integrated on a element (size dx): the smallest the element is, the more accurate is your solution. The extreme case where dx->0 will make converge the weak formulation to the strong one.
Establishing a weak formulation requires 3 steps: it "simply" consists in multiplying the strong form by any test-function and to proceed to the integration all over the domain: that makes two successive weakening. If your strong form is based on mechanics (or thermics) it may be interpreted as the global energy in the domain for any displacement test-function (or temperature). Solving it just means to find the displacement (or temperature) field that minimizes the weak form.
I also see two other interests for a weak form:
-1- an integration by part (third weakening) will reduce the highest derivative order ;
-2- ... and will make naturally appear your boundary conditions, some being called natural (just inject it in the boundary integral) some essential (need special treatment).
An interesting consequence is that the FEM approach that will discretize it, will necessarily be conservative! Indeed, flux conservation (I mean flux from #1 to #2 is opposite to flux from #2 to #1) along a common internal segment between two elements (#1 and #2) has been imposed (both integral terms vanish).
Dear Abdullah, to catch the importance of weak formulation in FEM it is better to refer a book by Jacob Fish and Ted Belytschko titled “A First Course in Finite Elements”. specially chapter three.
I think that the main reason is the lack of smoothness of the solution.
The weak forms actually require weak continuity, that's why we call them weak. The weak form differential equation is equivalent to the differential equation plus its boundary conditions of the strong form.
Because computers are not brains (on a lighter note). But if you are tending the mesh/steps to infinity the weak solution anyways become stronger!!
In the following another demonstration of the accuracy and stability of strong formulation. The results are documented by the following paper.
A comparison between Weak FEM and Strong FEM has been also presented.
Classical linear and parabolic finite elements are considered.
The C0 and C1 compatibility conditions have been implemented for the weak formulation.
The C1 compatibility conditions have been considered for the strong formulation. Spectral elements have been implemented too.
Convergence, accuracy and stability of the strong and weak methods are reported and compared.
Article The Strong Formulation Finite Element Method: Stability and Accuracy
The continuity requirements for the trial functions are relaxing in the FEM functional. If classical solution exists then it coincides to the smooth or weak solution
In the following another paper regarding the accuracy and stability of strong formulation.
Article A Strong Formulation Finite Element Method (SFEM) Based on R...
@Jibin Raj
Yes, there is you should write the weak form of the differential problem you want to solve (at least this works for structural mechanics)
Sir can you please explain the procedure for a 2-D Laplace equation. I am trying to apply Ritz method to solve. So first I need to get the weak form. In structural dynamics when dealing with strain energy relation we can apply the Ritz method directly. I need to do the same with Laplace equation.
Dear Jibrin Raj
Consider Laplace equation uxx + uyy = 0 ----------- (1)
The solution domain is discretized into a number of finite elements
that we shall assume to be the simple 3-noded triangular elements with
each element having local node numbers 1, 2, 3..
The potential u is assumed to vary across each element in accordance
with u = NUe = N1Ui + N2U2 + N3U3 ------------------------------(2)
where N = [ N1 N2 N3], Ue = [U1 U2 U3 ]
N = row vector of shape functions; Ue = column vector of nodal values of u
for an element.
A weak Galerkin formulation is obtained by multiplying the PDE by N and
integrating over each element to obtain
∫∫( uxx + uyy)Ndxdy = 0 ------------------------------ (3a)
Substituting u = NUe into Eq. (3a) we obtain
∫∫(NxxTN+ NyyTN)Uedxdy = 0 ------------------------ (3b)
where the superscript T stands for transpose and it has been introduced to
ensure correct matrix multiplication.
When Eq. (3b) is integrated by parts and line integrals are neglected it is
obtained that ∫∫(NxTNx+ NyTNy)Uedxdy = 0 ------------------------ (4)
Eq. (4) can be written simply as
KeUe = 0 (5)
because Ue is independent of x and y.
where Ke = an element 'stiffness' matrix = ∫∫ NxTNx + NyTNy dxdy with the integration being carried out over the element.
By summing equations such as Eq. (5) over all elements in the solution
domain we obtain a set of algebraic equations:
KU = 0 ...............(6)
where, K, the global 'stiffness' matrix is obtained by assembling all the
element 'stiffness' matrices Ke appropriately. U is the global vector for all nodal values of u. The term 'stiffness' arises from structural mechanics applications of the FEM. When a weak formulation is used only Co continuity can be enforced. It is not possible to obtain C1 continuity at the nodes, .i.e. the continuity of the 1st derivatives of u.
Note that the integral functional corresponding to Eq. 1 is
∫∫( ux2 + uy2) dxdy = 0 ................(7)
and this functional is used in applying the Ritz method over the whole solution domain.
If u = NUe is substituted in Eq.7 it is seen that Eq. 3b is reproduced readily.
This provides a strong proof of the validity of the Galerkin weak formulation.
It can be shown that for a 3-noded triangular element with nodes (x1, y1)
(x2, y2), (x3, y3)
Ni = (ai + bix + ciy)/(2Δ) ------------------------(8)
where Δ = area of triangle = a1 + a2 + a3
a1 = x2y3-x3y2, b1= y2-y3, c1 = x3- x2
the rest of the coeffs a2,a3, b2, b3, c2, c3 can be derived by cyclic permutation
of the subscripts 1, 2, 3.
Finally, the element stiffness matrix can be deduced to be
Ke = (1/4/Δ)* [ b12+ c12, b1b2 +c1c2, b1b3+c1c3;
b1b2 +c1c2, b22+ c22, b2b3 + c2c3;
b1b3 +c1c3, b2b3+ c2c3, b32 + c32 ]
using Matlab's notation. To obtain the global matrix K you need to know
how to post an element matrix Ke to the matrix K.
in my opinion the continuity or the regularity level of the interpolation space is the key to obtain high accuracy in the machanical formulation in strong or weak way. I think that the question is the continuity level of the interpolation as stated by F. Tornabene.
for example the blending property of the B-Spline interpolations is the trick for the high accuracy in the computational result. But high continuty means high sensibility to locking pathologies!!!
this it my opinion!
In the following another paper regarding the accuracy and stability of strong formulation.
This article represents a review and a survey regarding the Strong Formulation Finite Element Method.
Article Strong Formulation Finite Element Method Based on Differenti...
Dear Abdullah Waseem
Weak formulations are used in finite element when it is difficult/impossible to enforce at least some parts of boundary conditions. This could be due to a number of reasons.
For example in a barrelling compression test, the velocity BC are known at the die interface and axis/plane of symmetry. However, along the sample's free surface, the velocity boundary condition is not know. To be accurate we, don't know the position of the free surface and therefore, the velocity cannot be specified there.
The more BC can be enforced, the closer the solutions becomes to a strong formulation.
A strong formulation is unique but there are many weak formulations and therefore they are approximate solutions and not unique.
Hope this helps.
regards
Shahin
https://caendkoelsch.wordpress.com/2018/05/24/what-are-strong-and-weak-forms-in-finite-element-analysis-fea-why-do-we-need-them/
Check this link to know the conceptual difference.
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