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# What is the difference between essential boundary conditions and natural boundary conditions?

What is the difference between essential boundary conditions and natural boundary conditions, and what is the difference between primary variables and secondary variables?

Hi
For a differential equation of order (2n) , the derivatives of variable (dependent) with order 0 to n-1 are called essential derivatives. Moreover the derivatives of order n to 2n-1 are called natural derivatives. Then if the condition on the boundary is for the first type derivative it is essential boundary condition. If the condition on the boundary is of the second type it is natural boundary condition.
that is all.

Essential boundary conditions are conditions that are imposed explicitly on the solution and natural boundary conditions are those that automatically will be satisfied after solution of the problem. In the case of Finite Element approximations, the essential conditions will be exactly satisfied but the natural conditions only up to the order of the method . In many cases, the essential conditions correspond to Dirichlet boundary conditions when the problem is written as a boundary value problem for a partial differential equation. The natural condition corresponds to a Neumann condition, a stress-free condition, or something similar, depending on the problem. However, there are cases that are not so clear cut , so to identify Dirichlet=essential and Neumann=natural is not really correct. For instance, Dirichlet conditions can be assigned as natural conditions using Nitsche’s method, and Neumann conditions can be transformed to essential conditions using so-called mixed methods.
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Essential boundary conditions are conditions that are imposed explicitly on the solution and natural boundary conditions are those that automatically will be satisfied after solution of the problem. In the case of Finite Element approximations, the essential conditions will be exactly satisfied but the natural conditions only up to the order of the method . In many cases, the essential conditions correspond to Dirichlet boundary conditions when the problem is written as a boundary value problem for a partial differential equation. The natural condition corresponds to a Neumann condition, a stress-free condition, or something similar, depending on the problem. However, there are cases that are not so clear cut , so to identify Dirichlet=essential and Neumann=natural is not really correct. For instance, Dirichlet conditions can be assigned as natural conditions using Nitsche’s method, and Neumann conditions can be transformed to essential conditions using so-called mixed methods.
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I agree with the answer of Dmitrii, I just want to add a precision, natural boundary conditions are neumann boundary conditions or Fourier boundary conditions (the normal derivative added to a constant times the unknown are fixed) for an elliptic partial differential equation . Nevertheless, for elliptic partial differential equations, it is not possible to assign a value to the derivative of the unknown on the boundary (or on the part of the boundary), but only for the normal derivative (the directional derivative in the ouward normal of the domain, which is called neumann condition, this appears clearly when a variational formulation of the PDE is taken).
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Roberto Bernetti
Ministero dell'Istruzione, dell'Università e della Ricerca, Ancona, Italy
Speaking from the point of view of Finite Element Method I would be more close to the Martin Berggren answer adding that essential boundary condition involves restraints "directly" to the shape functions parameters, hence defining the subset in which you will search for the solution.
While natural boundary condition results as a by-product of the minimization process of the approximated functional used in the Finite Element Method in the subset previously defined.
I agree with Dmitri L, but some cases concern essential boundary conditions, which are built into the solution space, and an other concern natural boundary conditions are built into the weak form.
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I agree with Martin. Actually, the issue of whether one condition is classified as essential or natural is somewhat tricky. It very much depends on the problem at hand as well as on the variational formulation used to model that problem. For example, in the elasticity problem, a traction condition (for the normal component of the stress) in a pure displacement formulation is a natural condition, i.e., it is satisfied automatically by the variational formulation. On the other hand, the same probem, described by a mixed-formulation, where both displacement and stresses are independent unknowns, the traction condition becomes an essential condition, i.e., it is enforced a priori in the function space where one seeks the solution.
I would say, in genaral, that an essential condition is a condition which is enforced directly in the trial space, whereas a natural condition is satisfied "for free" through the variational formulaton of the problem.
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The essential boundary conditions are imposed on the functions in the space where the minimzation of the energy functional is made or the weak formulation is posed. For the standard FE formulation for the Poisson equation these are Dirischlet boundary conditions. However, if one uses the dual mixed formulation, then the
Dirichlet data become essentail while the Neumann BC are imposed on the flux variable which now is an independent unknown variable, this excatly what Stefano Micheletti is saying. Practically, to determine whether a BC is natural or essential,
one can do the following: To get the weak form you do some integration by
parts (or Stokes theorem); during this integration by parts process you need to deal with the boundary terms; then: (a) if you use the BC to produce some new terms in the bilinear and linear forms, these are natural BC; (b) if you need to elliminated them from the forms, you impose them on the functions of the space and then these are essential.
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For a differential equation of order 2n the conditions of order 0 to n-1 are essential and the conditions of order n to 2n-1 are natural.
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I agree with the answer of Martin. You can find more details in: "B.D. Reddy, Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements, Springer, 1998" .
Dear Abdullah,
Essential boundary condition= Dirichlet boundary condition,
Natural boundary condition = Neumann boundary condition
Thanks,
Murat,
And what about Robin Boundary conditions?
Of course, what I said is in Mathematics.
In other words, I agree with Dmitrii.
Dear Nicolas,
Actually all these BCs can be found in many books. I think it is not big deal, thanks.
Also, use Fuzzy membership function where uncertain???
The essential boundary conditions is the conditions concerning geometry, i.e. displacements, slope. etc. but the natural boundary conditions is the one concerning the force
You meant stress/strain (internal) type forces--not the external force(s), that will be then essential BC.
Dear Murat,
I know how to impose boundary conditions. But according to the "definition" in which group you would place this type. We can classify the BC's as
1 . Geometric BC
2. Natural BC
Geometric BC's are due to the geometry of the problem(Displacement or Slopes)
Natural BC's are due to the force (forces or stresses) on the problem.
For example,
In a Fixed Free string of length 'L',( i.e L.H.S fixed, R.H.S free) BC of L.H.S is W(0,t)=0 is essential BC. R.H.S BC is Derivative of W(x,t) at 'L' w.r.to 'x' is Natural.
I do not agree that "geometric b.c."="essential b.c." and that "b.c. related to forces"="natural b.c.". This may be mostly the case, but the question if a boundary condition is related to "geometry" or "forces" is a physical question wheras the question if a b.c. is an "essential" or a "natural" one is pure mathematical question.
In physical notation boundary conditions related to geometrical considerations are called "kinematic boundary conditions" and those who are based on stress/strain of forces are called "dynamic boundary conditions".
In contrast to this, in mathematics a boundary condition is called "essential" if it is part of the original formulation of the problem from the very beginning, as already stated by Martin. The natural bondary conditions, however, are an outcome from the variational formulation your FEM is based on: If you do a FE calculation with one or several boundary conditions missing, you allow variation of your fields at the boundary which automatically implies additional conditions. These are due to the variational calculus and are called "natural boundary conditions".
There are many famous examples, where the essential b.c. are of Dirichlet type and related to geometry at the same time, whereas natural b.c. are of Neumann type and are related to forces or stresses, however, this is not always the case!
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Essential boundary conditions involve deflection and slope while natural boundary conditions involve bending moments and shear forces.
your question seems to have stemmed from FEA. Nevertheless these boundary conditions are best FELT if one takes recourse to the beginning i.e the calculus of variation(FEA is closely related to Rayleigh method which is practically the approximate technique exploited from variational calculus)..
While extremizing a functional involving derivatives explicitly,say highest of kth derivative (e.g for a beam the relevant functional is total potential energy containing 4th order derivative of deflection v say) the intergration by parts yields an integral and the limit terms with variation of derivatives of v upto k-1 ,each multiplied with a certain function .If the values of derivatives of v are prescribed at the boundary then the variations will be zero,if they are not prescribed then that certain function it is multiplied with ll be zero.the 1st case is called a rigid or kinematic boundary condition and the 2nd case these are called natural b.c. ,they are naturally satisfied if the euler lagrange equation is to yield an extremizing function(in the beam case the equilibrium equations yielding displacements);
primary variable is the one which appears in the variational formulation(in the displacement based FE displacements are calculated from K d=R) the secondary variables are stress which is calculated from displacements employing the suitable constitutive law.
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Marcelo, I know these BC and know how to impose in a FEM code. But I was talking about the classification in Essential and Natural, and the correspondence to Dirichlet and Neuman.
Another case for Essential and Natural is presented in this paper http://www.imechanica.org/files/blochfem-accepted.pdf, where the Essential BCs are the Natural BCs are related to Bloch theorem.
In general, essential boundary conditions are conditions imposed to the variables (i.e. related to the degrees of freedom such as displacements, temperature, pressure, etc.), while natural conditions are those imposed to the loads. For example in the system Ku=f; essential conditions are imposed to u while natural conditions are considered in the vector f.
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To get respose to the question about the classification in Essential and Natural, and the correspondence to Dirichlet and Neuman, see my thesis.
In Finite element method essential boundary conditions are the boundary conditions which is specifically defined in the strong form of the variable "u", while defining the weighting function to derive the weak form, it should be made sure that the weighting function is zero at the essential boundary condition and hence enforce any function multiplying this function to zero at the said boundary condition. The natural boundary condition on the other hand projects itself automatically from the strong form and into the weak form without having to define it especially.
IF a boundary condition involves one or more variables in a 'direct' way it is essential otherwise it is natural.
For a differential equation of order 2n the conditions of order 0 to n-1 are essential and the conditions of order n to 2n-1 are natural.
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Although abstract generalizations (as stated in many answers) are good, often specific examples and simplifications do help in understanding. With this view I am commenting as follows:
In the context of FEM, natural boundary conditions are those which are automatically satisfied by the solution, such as adiabatic boundary in heat conduction analysis and force free surfaces in stress analysis. (Although, sometimes specifying convective heat loss from surface is also included in natural boundary condition -these are Neumann boundary conditions).
Specifying primary variable of the problem - temperature in heat conduction and displacement in stress analysis - is called essential boundary condition (Dirichlet boundary condition).
Specifying secondary variable of the problem - convective heat loss from surface or force at boundary in stress analysis - is also natural boundary condition (Neumann boundary condition) (See book FEM by J.N.Reddy). Although I would prefer to call it as Neumann boundary condition. and leave those conditions, where these secondary variables are zero at boundary, as natural boundary conditions because these are naturally satisfied by solution without imposing any conditions from outside.
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Lyazid Bouhala
Luxembourg Institute of Science and Technology (LIST)
Hi,
Essential BC concern the variables (the field) and Natural BC concern the derivatives of the variables (the flux). Prescribed displacements are Ess. BC and prescribed forces are Nat. BC. They are also called respectively Dirichlet and Newman BC.
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Hi
Essential boundary conditions are the ones explicitly imposed on the solution while natural boundary conditions are the ones that are consequently be satisfied after problem solution is achieved.
Regards
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In the finite element formulation one usually writes week formulation using the process of integration by parts, to lessen the smoothness demand on the numerical solution. Neumann boundary conditions appear naturally in surface integral terms of the week formulation, thus they are sometimes called natural boundary conditions, as opposed to the Dirichlet boundary conditions that are called essential boundary conditions.
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In the above answer, "week" should be "weak",. Sorry for the typo.
Force/ natural boundray condition present a some types of constraints placed on the boundary with respect to magnitude of forces.
Geometric/ Essential/ Rigid boundary condition place constraints on the magnitude of displacement & slope at given any position on the boundary.
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The notion of essential versus normal (boundary conditions) does not appear for general elliptic partial differential equations with boundary conditions, as can be checked by looking at the famous work (in the late 1950s) of Shmuel AGMON, Avron DOUGLIS (1918-1985), and Louis NIRENBERG.
The notion appears in the smaller class of variational problems, and is related to the so-called Dirichlet principle'': RIEMANN (1826-1866) heard it from DIRICHLET (1805-1859), so that he named it after him, but GAUSS (1777-1855) and GREEN (1793-1841) had made the observation before; THOMSON (1824-1907) made the same observation later, but since in 1892 he was made baron Kelvin of Largs, and thereafter known as Lord Kelvin, one finds mention of a Kelvin principle or a Thomson principle.
The idea is that solving -\Delta u = f with Dirichlet conditions is related to minimizing an energy \int_{\Omega} [|grad(u)|^{2} - 2 f u] dx, and WEIERSTRASS (1815-1897) observed that it is not clear that the functional attains its minimum in C^{2} (where the equation has a classical meaning) or C^{1} (where the functional is defined) and this led him to define and study the notion of compactness. F. RIESZ (1880-1956) showed afterward that the unit ball of an infinite dimensional space is never compact, and the natural function space'', denoted H^{1}, was later introduced by Sergei SOBOLEV (1908-1989), and compact then needs to be understood for a weak topology.
Here I use natural as a way to say from a physical point of view'': the equation appears in electrostatics, a simplification of the system of equation by MAXWELL (1831-1879), for which one actually uses the much simpler formulation by HEAVISIDE (1850-1925), and the Sobolev space H^{1} corresponds to a finite electrostatic energy (i.e. the electromagnetic energy with the magnetic field H = 0).
Then, the condition named after NEUMANN (1798-1895) does not make sense on H^{1}, since the derivatives belong to L^{2}, hence cannot be restricted on a set of Lebesgue measure 0. However, if the minimum is attained on H^{1} (which requires \int_{\Omega} f dx = 0) at a function belonging to H^{2} (for example), the Neumann condition makes sense.
The condition named after FOURIER (1768-1830) or ROBIN (1855-1897) appears if one adds to the functional c\int_{\partial \Omega} u^{2} d\sigma.
When I started studying with Jacques-Louis LIONS (1928-2001) in the late 1960s, he taught that (without the historical perspective), but he had also been working with Enrico MAGENES (1923-2010) on questions like if u \in H^{1} and \Delta u \in L^{2}, then the normal derivative (at the boundary) exists, in a negative Sobolev space for the boundary''; their method extends to variable but smooth coefficients.
Then Jacques-Louis LIONS observed (and taught) about the functional space H(div;\Omega) [i.e. u_{1},...,u_{N}\in L^{2}(\Omega) and div(u)\in L^{2}(\Omega)] for which the normal trace exists, and his method extends to variable non-smooth coefficients.
This point of view permitted to put into a mathematical framework the mixed methods which engineers had introduced for (linearized) elasticity, which were natural from their point of view: the approximation of elastic behaviour is only valid if the stresses stay below some critical level, so that one needs to be precise on stress, while the mathematicians thought that one had to be precise on displacements.
The essential boundary conditions'' make sense on the functional space used, while the normal
boundary conditions'' do not make sense on the functional space used, but since the functional considered attains it minimum at a slightly more regular function, it then makes sense.
In other words, the qualifiers essential / normal are not related to the PDE with its associated boundary conditions, but to the variational formulation chosen to make all these appear. Then, if one knows more than one variational formulation for the same PDE + boundary conditions, the notion may become fuzzy.
Best regards
Luc TARTAR
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From mathematical point of view two general boundary conditions may be considered. Dirichlet or essential B.C.'s versus Neumann or natural or free B.C.'s. The 1st one refer to the primitive variable of the problem when the second refer to the so-called secondary variables. these two type of B.C.'s are equivalent in the exact solution of a boundary valued problems. The basic difference emerges in numerical FEM solution. Drichlet B.C.'s have to be specified explicitly whereas Neumann or natural or free conditions are dealt with implicitly as part of the formulation.
Spectral/hp element Method for cfd, Karniadakis and Shervvin, Ch. 2
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In elliptic P.D.E ,the boundary conditions are those assuring the V-ellipticicity of the bilinear functional assuring the positive definiteness of the stiffness matrix (in the standard finite element formulation). Those boundary conditions will be used explicitly in the solution. they are called essential. For more details consult the book of Ciarlet and that of Zienkiewicz
Abbas AbdulMajeed Allawi
University of Baghdad / College of Engineering/ Department of Civil Engineering
Thanks Luc Tartar  for this clarifying.
Hello
It is possible to discrete a differential equation as n linear equations in terms of n essential derivatives and n natural derivatives at n nodes. when n out of 2n parameters are known one may solve for the n unknown parameters. For a differential equation of order 2n the derivatives of orders 0 to n-1 are called essential derivative and the derivatives of order n to 2n-1 are called natural derivatives.
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For an elliptic PDE of order 2m we use Green theorem to weakening  the required smoothness in the original formulation of the problem (variational formulation). In this case the higher order boundary conditions (Natural BCs) simply handle in the formulation. However, the lower order BCs ( of order less than m) can't be handled in the formulation and one have to set these conditions  to the solution space, i.e., the basis function originally should satisfy these conditions.
for further information see: Daya Reddy, Introductory Functional Analysis. With Applications to Boundary, Springer, 1998.
Mohsen Rezaee Hajidehi
Institute of Fundamental Technological Research
Generally, essential boundary conditions are the ones that are imposed on the degree of freedom itself (e.g. displacement). On the other hand, natural boundary conditions are applied on the derivates of the degree of freedom (e.g. velocity).
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Abbas AbdulMajeed Allawi
University of Baghdad / College of Engineering/ Department of Civil Engineering
Dear Abdullah Waseem
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attached reference may be beneficial for you..regards
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Abbas AbdulMajeed Allawi
University of Baghdad / College of Engineering/ Department of Civil Engineering
It is a good reference DR Amjad
Hi
In the context of numerical analysis of differential equations the notion of essential and natural derivatives arises. For a differential equation of order (2n) of the function y (i.e. (y(n))(n)+...=0) the derivatives of the order y(m), m=0,n-1 are called the essential, and the derivatives of the order y(m) , m=n, 2n-1 are called the natural. The values of these derivatives on the boundary are called the boundary conditions. Any numerical method is a relation between these two derivatives. For example in structural mechanics the natural derivatives have the force character and the essential derivatives have the displacement character, therefore a numerical method is a force- displacement relation. Based on the notion of these derivatives a consistent relation between the differential equations and the numerical method concepts may be obtained. For example the concept of the stiffness may be defined in the numerical domain and at the same time in the differential equation domain.
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Hail,
Generally, the applicable problems in engineering categorized to three kinds: 1- Initial Value Problem, 2- Boundary Condition Problem, and 3- Eigen Value Problem.
In the second kind of problems, two type of boundary condition have been presented: (a) Essential or Geometrical Boundary Condition, and (b) Natural Boundary Condition.
In a simple support beam with length ‘L’ two type of boundary condition (a, b) has been described as follows: (a) Essential or Geometrical Boundary Condition means displacement in the lengths x=0 and x=L in vertical direction are zero and (b) Natural Boundary Condition means the value of bending moment force in the lengths x=0 and x=L in rotational direction are zero.
(a) Essential or Geometrical Boundary Condition
Y(x=0) =0 and Y(x=L) =0
(b) Natural Boundary Condition
M(x=0) =0 and M(x=L) =0
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Finally, in this type of Problems the value of redundant of system based on the behavior of system and the constraint of system have been defined. In other word the geometry of structure or system of problem have significantly role in order to approaching of analysis and solution of problem. Consequently, for description of Boundary Condition Problems could be say that the problem from one side is unknown and from other side in the all constraint during all time of problem the variable including ‘Essential or Geometrical Boundary Condition’ and ‘Natural Boundary Condition’ are determine.
Good Luck,
I think one would like to refer to a reference about the terminology of the finite element modelling. The reference in the given link may be useful: https://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/IFEM.Ch06.d/IFEM.Ch06.pdf
Best wishes
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Hi
The notion of the essential and natural derivatives/boundary condition, are definitions made in mathematics. The former defined as displacement and the latter force. In the context of numerical methods the former is the multiple of stiffness and is conventionally located on the left side of equation and the latter is located on the right side of equation.
That is all!
Yuwaraj Marotrao Ghugal
Government Engineering College, Karad, Maharashtra State, India
I suggest readers to Refer book on Solid Mechanics by Dym and Shames: Variational Approach for more insight on boundary conditions resulted out of extremization process of total energy functional and application of fundamental lemma of calculus of variations.
Dr Ghugal YM
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Hi
For a differential equation of order (2n) , the derivatives of variable (dependent) with order 0 to n-1 are called essential derivatives. Moreover the derivatives of order n to 2n-1 are called natural derivatives. Then if the condition on the boundary is for the first type derivative it is essential boundary condition. If the condition on the boundary is of the second type it is natural boundary condition.
that is all.

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