[email protected] , 上海交通大学›º体力学课件/class-9...na26018 finite element analysis...

2020年

万德成[email protected] , http://dcwan.sjtu.edu.cn/

上海交通大学船舶海洋与建筑工程学院海洋工程国家重点实验室

Class-9

NA26018

Finite Element Analysis of

Solids and Fluids

NA26018 Finite Element Analysis of Solids and Fluids

Contents

Equations of Fluid Dynamics

Convection-Dominated problems Brief introduction for Petrov and Charactrised based FE

method

CBS Algorithm: A General Procedure for Compressible and Incompressible Flow

SUPG/PSPG Method for Incompressible Flow



Constitutive relations for fluids

The stress-strain rate relations for a linear (Newtonian) isotropic fluid require the definition of two constants. The first of these

links the deviatoric stresses 𝜏𝑖𝑗 to the deviatoric strain rates

In the above equation the quantity in brackets is known as the

deviatoric strain rate, δ𝑖𝑗 is the Kronecker delta

휀𝑖𝑗 is rates of strain, the primary cause of the general stresses 𝜎𝑖𝑗.they are defined in a manner analogous to that of infinitesimal strain in solid mechanics as



without necessarily implying incompressibility 휀𝑘𝑘 = 0

The above relationships are identical to those of isotropic linear elasticity as we will note again later for incompressible flow

However, in solid mechanics we often consider anisotropic materials where a larger number of parameters are required to define the stress-strain relations

Nonlinearity of some fluid flows is observed with a coefficient 𝜇depending on strain rates and/or other variables such as temperature. We shall term such flows “non-Newtonian”



Conservation principles – 1. Mass conservation

If 𝜌 is the fluid density then the balance of mass flow 𝜌𝑢𝒊 entering and leaving an infinitesimal control volume is equal to the rate of change in density as expressed by the relation

is known as the gradient operator

NOTE: In this section and subsequent ones, flow in the control volume (fixed

in space) is research object. This is known as the “Eulerian form”. It is different in contrast to the usual treatment in solid mechanics where displacement is a primary dependent variable

It is possible to recast the above equations in relation to a moving frame of reference and, if the motion follows the particle, the equations will be named “Lagrangian”



Conservation principles – 2. Momentum conservation

In the jth direction the balance of linear momentum leaving and entering the control volume is to be in dynamic equilibrium with

stresses 𝜎𝑖𝑗 and body forces 𝜌𝑔𝑗. This gives a typical component equation

Or by using the essential constitutive relation,



Conservation principles – 3. Energy conservation

In momentum and mass conservation equations, the dependent variables are 𝑢𝑖, 𝑝 and 𝜌

The deviatoric stresses are defined in terms of velocities and hence are dependent variables

Obviously, there is one variable too many for this equation system to be capable of solution

If the density is assumed constant (as in incompressible fluids) or if a single relationship linking pressure and density can be established (as in isothermal flow with small compressibility) the system becomes complete and solvable

For an ideal gas this takes the form：



In a general case, it is necessary to supplement the governing equation system by equation of energy conservation.

The balance of energy in an infinitesimal control volume can be written as

• Total energy per unit mass

• Intrinsic energy per unit mass(dependent on the state of the fluid)

• Enthalpy

• Conductive heat flux for isotropic material


The steady-state problem in one dimension

Convection-Dominated problems

Consider the discretization of

with

where 𝑁𝑎 are shape functions and 𝝓 represents a set of still unknown parameters. Here we shall take these to be the nodal values of 𝜙

A linear shape function for a one-dimensional problem



where

For a typical internal node a the approximating equation becomes

The weighted residual form of the one-dimensional problem is written as

Integrating the second term by parts gives



For linear shape functions, Galerkin weighting (𝑊𝑎 = 𝑁𝑎), and elements of equal size ℎ, we have for constant values of 𝑈, 𝑘, and 𝑄

where

and the domain of the problem is 0 ≤ x ≤ L



yields a typical assembled equation (after multiplying by ℎ/𝑘) for an inside node “a”

where

is the element Peclet number For the case of constant 𝑄 the above is identical to the usual

central finite difference approximation obtained by putting



Approximations to 𝑈𝑑𝜑/𝑑𝑥 – 𝑘𝑑2𝜑/𝑑𝑥2 = 0 for 𝜑(0) = 1 and 𝜑(𝐿) = 0 for various Peclet

numbers.

Solid line: exact solution; dotted line with triangular: standard Galerkin solution



The algebraic equations are obviously non-symmetric and their accuracy deteriorates as the parameter 𝑃𝑒 increases

Indeed as 𝑃𝑒 → ∞ (i.e., when convective terms are of primary importance), the solution is purely oscillatory and bears no relation to the underlying problem

It is easy to show that with the standard Galerkin procedure oscillations occur when

|𝑃𝑒| > 1

Summary

The above example clearly demonstrates that the standard Galerkin method in which 𝑊𝑎 = 𝑁𝑎 cannot be used to solve problems in which convective terms are large compared with those of diffusion



Motivated by the fact that the propagation of information is in the direction of the velocity 𝑈

finite difference practitioners were the first to overcome the bad approximation problem of the central difference method

One-sided finite differences is used to approximate the first derivative

the Upwind finite difference form of the governing equation approximation:

Central finite difference approximation:



NEXT: How can upwind differencing be

introduced into a finite element scheme?

How to generalized more complex situations?

With this upwind difference approximation (--□- -), nonoscillatory solutions are obtained through the whole range of Peclet numbers

Exact nodal solutions are obtained for pure diffusion (𝑃𝑒 = 0) and for pure convection (𝑃𝑒 = ∞); however, results for other values of Pe are generally not accurate



Petrov-Galerkin methods for upwinding in 1-dimension

The first possibility is that of the use of a Petrov-Galerkin type of weighting in which 𝑊𝒂 ≠ 𝑁𝑎. Such weighting was first suggested by Zienkiewicz et al. (1976). For elements with linear shape functions 𝑁𝑎, we shall take weighting functions 𝑊𝑎 so that

𝑊𝒂∗ is obtained for finite difference equivalent

Various forms of 𝑊𝒂 are possible, the most convenient is the following simple definition which is a discontinuous function

With the above weighting functions (𝑊𝑎), the approximation for a typical node 𝑎 becomes



𝑄 is assumed constant for the whole domain and equal length elements are used

Discussion for 𝛼 with 𝛼 = 0 the standard Galerkin approximation is recovered with 𝛼 = 1 the full upwind form is available if the value of 𝛼 is chosen as

then exact nodal values will be given for all values of 𝑃𝑒（ see Fig. in last page ）

coth: hyperbolic cotangent function



Although the proof of 𝛼opt was given for the case of constant coefficients and constant size elements

Nodally exact values will also be given if 𝛼 = 𝛼opt is chosen for each element individually

Variable source term with variable element sizes

Variable source term/U with variable element sizes



Continuity requirements for weighting functions

The weighting function 𝑊𝑎 (or 𝑊𝑎∗) can be discontinuous as far as

the contributions to the convective terms are concernedRecall the FE Eq. for steady-state convection-diffusion problem:

where

Clearly no difficulty arises at the discontinuity in the evaluation of the integrals for advection phase

However, when evaluating the diffusion term, a local infinitywill occur with discontinuous 𝑊𝑎



To avoid this difficulty we mollify or smooth the discontinuity of the 𝑊𝑎 so that this occurs within the element and thus avoid the discontinuity at the node

Petrov-Galerkin weight function 𝑊𝑎



Balancing diffusion in one dimension

The comparison of the nodal obtained on a uniform mesh and for a constant 𝑄 shows that the effect of using the Petrov-Galerkinprocedure is equivalent to the use of a standard Galerkin process with the addition of a diffusion

Standard Galerkin process:

Petrov-Galerkin procedure:



Discussion:

Such balancing diffusion is easier to implement than Petrov-Galerkin weighting, particularly in two or three dimensions, and has some physical merit in the interpretation of the Petrov-Galerkin methods

However, balancing diffusion does not provide the required modification of source terms (not suitable for variable 𝑄)

The concept of artificial diffusion introduced frequently in finite difference models suffers from the same drawbacks and in addition cannot be logically justified



The steady-state problem in two (or three) dimensions

The equation now considered is steady-state version in multidimensions, i.e.,

Streamline (upwind) Petrov-Galerkin weighting (SUPG) :

The most obvious procedure is to use again some form of Petrov-Galerkin method, seeking optimality of 𝛼 in some heuristic manner. The Peclet parameter

is now a “vector” quantity, hence that upwinding needs to be “directional”



In two (or three) dimensions the convection is only active in the direction of the resultant element velocity 𝑈, hence the diffusion introduced by upwinding should be anisotropic, only in the direction of the resultant velocity

where

suppose that the velocity components 𝑈𝒊 in a particular element are constant and that the element size ℎ can be reasonably defined e.g.:



Introducing the approximation

Using the weight

SUPG method is computed from the weighted residual form

Integration by parts and introduction the natural boundary condition

where



In the discretized form the “balancing diffusion” term becomes

This indicates a zero coefficient normal to the convective velocity vector direction. It is therefore named streamline upwind Petrov-Galerkin (SUPG) process

The streamline diffusion should allow discontinuities in the direction normal to the streamline to travel without appreciable distortion

However, with the standard finite element approximations actual discontinuities cannot be modeled and in practice some oscillations may develop

For this reason some investigators add a smoothing diffusion in the direction normal to the streamlines (crosswind diffusion)



Transients problems

To allow a simple interpretation of the various methods and of behavior patterns, the scalar equation in one dimension in nonconservation form is considered

The problem so defined is nonlinear unless 𝑈 is independent of 𝜑

The main behavior patterns can be determined by a change of the independent variable 𝑥 to 𝑥′ such that

Noting that for 𝜑 = 𝜑(𝑥’ , 𝑡)we have

The coordinate system describes characteristic directions and the movingnature of the coordinates must be noted



The one-dimensional equation now becomes simply

equations of this type can be readily discretized with self-adjoint spatial operators and solved by standard finite element Galerkin procedures

A further corollary of the coordinate change is that with no conduction or source terms, i.e., when 𝑘 = 0 and 𝑄 = 0, we have simply

or

along a characteristic (assuming 𝑈 to be constant). This is a typical equation of a wave propagating with a velocity 𝑈 in the 𝑥direction



Characteristic-based methods

We have already observed that, if the spatial coordinate is “convected” in the manner implied, i.e., along the problem characteristics, then the convective, first-order, terms disappear



The remaining problem is that of simple diffusion for which discretization procedures with the standard Galerkin spatial approximation are optimal (in an energy norm sense)

The most obvious use of this in the finite element context is to update the position of the mesh points in an incremental Lagrangian manner

On the updated mesh only the time-dependent diffusion problem needs to be solved using the Galerkin method



It seems obvious that after completion of a single step a return to the original mesh should be made by interpolating from the updated values to the original mesh positions

The process of continuously updating the mesh and solving the diffusion problem on the new mesh is, of course, impractical.

When applied to two- or three-dimensional configurations very distorted elements would result and difficulties will always arise near the boundaries of the domain

The method described is somewhat intuitive but has been used with success for solution of transport equations by Adey and Brebbia and others as early as 1974.

The procedure can be formalized and presented more generally and gives the basis of so-called characteristic-Galerkin methods



Characteristic-Galerkin proceduresWe shall consider that the equation of convective diffusion in its one-dimensional form is split into two parts such that

and separate the differential equation into two additive parts. A convective part:

And the self-adjoint terms [𝑄 contains the source, reaction, and term (𝜕𝑈/𝜕𝑥)𝜙]

Both 𝜙∗and 𝜙∗∗ are approximated by expansions



Standard Galerkin discretization of the diffusion equation allows 𝝓∗∗𝑛+1 to be determined by solving an equation of the form

with



In solving the convective problem we assume that 𝜙∗ remains unchanged along the characteristic. The new value is given by

As we require 𝝓∗𝑛+1 to be approximated by standard shape functions, we shall write a projection for smoothing of these values as

giving

Above integrals is still complex, especially if the procedure is extended to two or three dimensions

The stability is dependent on the accuracy of such integration The scheme is stable and exact as far as the convective terms are

concerned if the integration is performed exactly However, stability and indeed accuracy will even then be affected by

the diffusion terms



A simple explicit characteristic-Galerkin procedure

Until now, all methods then proposed are somewhat complex in programming and are time consuming For this reason a simpler alternative (explicit characteristic-

Galerkin method) was developed in which the difficulties are avoided at the expense of conditional stability

Its derivation involves a local Taylor expansion

Recall Eq. along the characteristic

In the moving coordinate 𝑥′, the convective acceleration term disappears and source and

diffusion terms are averaged quantities along the characteristic Now the equation is self-adjoint and the Galerkin spatial

approximation is optimal



The time discretization of the above equation along the characteristic gives

where 𝜃 = 0 for explicit forms, and between 0 and 1 for semi-implicit and fully implicit forms The solution of the above equation in moving coordinates leads

to mesh updating and presents difficulties, so we will suggest alternatives

From the Taylor expansion we have

(a)

(b)



assuming 𝜃 = 0.5

where 𝛿 is the distance traveled by a particle in the x-direction

where 𝑈 is an average value of 𝑈 along the characteristic

(c)



Different approximations of 𝑈 lead to different stabilizing terms. The following relation is commonly used

Consider the later approximation of 𝑈 and using the Taylor expansion,

Inserting (b) (c) to (a), with 𝜃 = 0.5, we have



In the above equation, higher-order terms are neglected If 𝑛 + 1/2 terms in the above equations are replaced with 𝑛

terms, the equations become explicit in time

where

where



For multidimensional problems, Eq. can be written for the fully explicit form as

Recall original Eq. is written along the characteristic, the Galerkinspatial approximation can be used



use the weighting 𝑁T in the integrated residual expression. Thus we obtain

𝐾𝒖 and 𝑓𝑠𝑛 are from the new term introduced by the discretization

along the characteristics



The characteristic-Galerkin algorithm could have been obtained by applying a Petrov-Galerkin weighting

to the various terms of the governing Eq. excluding the time derivative 𝜕𝜙/𝜕𝑡 to which the standard Galerkin weighting of 𝑁𝑇 is attached



A two-dimensional application of the characteristic-Galerkinprocess is illustrated in which we show pure convection of a disturbance in a circulating flow

(a) original form; (b) form after one revolution using consistent M matrix; and (c) form after one revolution using lumped mass (Lax-Wendroff)

Present scheme is contrasted with the solution obtained by the finite difference scheme of Lax and Wendroff which for a regular mesh gives a scheme identical to the characteristic-Galerkinexcept for the mass matrix, which is always diagonal (lumped) in the finite difference scheme

The difference is entirely due to the proper form of the mass matrix 𝐌

For transient response the importance of the consistent mass matrix is crucial. However, the numerical convenience of using the lumped form is overwhelming in an explicit scheme



It is easy to recover the performance of the consistent mass matrix by using a simple iteration

with 𝑆𝑛 being the right-hand side of Eq., and

Substituting a lumped mass matrix 𝐌𝐿 to ease the solution process we can iterate as follows:

𝑙 is the iteration number. The process converges very rapidly



Nonlinear waves and shocks

The procedures developed in the previous sections are in principle available for both linear and nonlinear problems (with explicit procedures of time stepping being particularly efficient for the latter). Quite generally the convective part of the equation, i.e.,

In the one-dimensional case with a scalar variable we shall have equations of the type



Progression of a wave with velocity 𝑈 = 𝜙 (Burger Eq.)

Different parts of the wave moving with velocities proportional to their amplitude cause it to steepen and finally develop into a shock form. This shock propagates at a finite speed (which here is the average of the two extreme values)

In such a shock the differential equation is no longer valid but the conservation integral is. We can thus write for a small length Δ𝑠around the discontinuity



where 𝐶 = limΔ𝑠

Δ𝑡is the speed of shock propagation and 𝜙 and 𝐹

are the discontinuities in 𝜙 and 𝐹 respectively. Eq. is known as the Rankine-Hugoniot condition

Discussion： Shocks develop frequently in the context of compressible gas

flow and shallow-water flow and can often exist even in the presence of diffusive terms in the equation

Approximation of the finite element kind in which we have postulated in general a 𝐶0 continuity to 𝜙 can smear such discontinuity over an element length, and generally oscillations near such a discontinuity arise

To overcome this problem artificial diffusivity is frequently used. This artificial diffusivity must have the following characteristics: 1. It must vanish as the element size tends to zero2. It must not affect substantially the smooth domain of the solution



Additional Lapidus-type diffusivity with different coefficient 𝐶𝑙𝑎𝑝

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