5. Formulation of the quantity of interest

As we already saw in the first part, instead of evaluating a measure of error such as the energy norm, for example, it is more useful to estimate the error on a quantity of interest that makes physical sense. This quantity of interest (quantity of interest or*output*) can be represented by a linear functional \(Q(\cdot )\) defined in the space of test functions. The purpose of estimating the quantity of interest is therefore to ensure the quality of \(Q({u}^{h})\) by estimating the quantity:

\[\]

: label: EQ-None

{varepsilon} ^ {Q} =Q (u- {u} ^ {h}) =Q (e)

To get the estimate, you must use functional \(Q(\cdot )\) as the loading of the dual problem defined by:

\[\]

: label: EQ-None

{begin {array} {cc}text {Find}text {Find}omegain Vtext {such as} &\ a (v,omega) =Q (v) &forall vin Vin Vend {array}

Numerous physical quantities are used in design and need to be controlled. These quantities can be, for example, the average of a component of the displacement or the average of a component of the stresses on a subdomain (area of interest):

\[\]

: label: EQ-None

Q (v) =frac {1} {midomegamid} {omegamid} {int} _ {omega} {v} _ {x} domega

\[\]

: label: EQ-None

Q (v) =frac {1} {midomegamid} {omegamid} {int} _ {omega} {sigma} _ {mathrm {xx}} domega

These quantities cannot be used directly to load the dual problem. They must be expressed appropriately in order to find the burden to impose on the local problem. The general form of these quantities is as follows [bib25]:

\[\]

: label: EQ-None

Q (v) = {int} _ {Omega} {f} {f} {f} ^ {f} ^ {f} {f}} {f} {Q}f} {Q}cdot vmathrm {dGamma} -a ({u} ^ {Q}, v)

where \({f}^{Q}\), \({F}^{Q}\), and \({u}^{Q}\) are given. In the following, we will see that simple quantities of interest are deduced directly from this general form to constitute particular cases in which one or more terms are zero.

5.1. Quantities associated with travel

5.1.1. Component of displacement in a domain

The first quantity related to displacement is the average value of a displacement component in subfield \(\omega\):

\[\]

: label: EQ-None

begin {array} {cc} {Q} ^ {text {depl}} (v) =frac {1} {midomegamid} {int} _ {v} _ {v} _ {i} _ {i} _ {i} domega} domega & i=mathrm {1,2}text {or} 3end {array} _ {v} _ {v} _ {v} _ {i} _ {i} domega} di} domega & i=mathrm {1,2}text {or} 3end {array}

\({Q}^{\text{depl}}(v)\) can be written as:

\[\]

: label: EQ-None

{int} _ {Omega} {f} {f} ^ {Q}cdot vdOmega

which corresponds to the particular case where \({F}^{Q}=0\) and \({u}^{Q}=0\) and where \({f}^{Q}\) is a constant vector on \(\omega\) and zero elsewhere.

The dual problem to be solved is written as follows:

\[\]

: label: EQ-None

begin {array} {cc} {int} _ {omega} {omega} {sigma} _ {mathrm {ij}} (v) {varepsilon} _ {mathrm {ij}} (omega) dint} _ {omega} {omega} {f} _ {i} _ {i} _ {i} domega &forall vin Vend {array}

Using the symmetry property of the stress tensor and deriving, we get:

\[\]

: label: EQ-None

{int} _ {omega} {({sigma}} _ {mathrm {ij}}} (omega) {v} _ {i})} _ {, j} dOmega - {int} _ {omega} _ {omega} {sigma} _ {sigma} _ {omega} _ {omega} _ {omega} _ {omega} _ {omega} _ {omega}} {f} _ {i} ^ {Q} {Q} {v} _ {i} domega

Finally using the Stokes formula, we have:

\[\]

: label: EQ-None

{int} _ {omega} ({sigma} _ {mathrm {ij}, j} (omega) + {f} ^ {Q}) {v} _ {i} dOmega - {int} _ {int} _ {partialOmega} _ {partialOmega} _ {partialOmega}} {partialOmega} _ {i} dOmega =0

This allows us to write the local dual problem:

\[\]

: label: EQ-None

{begin {array} {cccc} {sigma} _ {sigma} _ {mathrm {ij}, j} (omega) + {f} _ {i} ^ {Q} &text {=} & 0&mathrm {in}mathrm {in} {in}}mathrm {in} {in}}mathrm {in}}mathrm {in}}mathrm {in}in}\ mathrm {in}in}\ omega{u} _ {i} (x) &text {=} & 0&mathrm {on} {mathrm {in} {in}}mathrm {in}} {in}mathrm {in}}mathrm {in} U}\ {sigma} _ {mathrm {ij}}} (omega) {n} _ {j} (x) &text {=} & 0&mathrm {on} {Gamma}} {Gamma} _ {F}end {array}

We thus note that imposing \({Q}^{\mathrm{depl}}(v)\) as the loading of the dual problem is the same as imposing a constant volume loading on \(\omega\). It should be noted that if we look for the error on the complete displacement vector, it will be necessary to solve as many dual problems as there are components (2 in dimension 2 and 3 in dimension 3).

5.1.2. Component of moving on an edge

Another useful quantity related to displacement is the average value of a component of displacement on an edge \(\gamma \subset {\Gamma }_{F}\):

\[\]

: label: EQ-None

begin {array} {cc} {Q} ^ {mathrm {depl}} (v) =frac {1} {midgammamid} {int} _ {gamma} {v} _ {v} _ {i} _ {i} _ {i} dmathrm {depl}}} (v) =frac {1}} {midgammamidgammamid} {int} _ {v} _ {v} _ {v} _ {v} _ {i} _ {i} _ {i} dgamma & i=mathrm {1.2}}mathrm {or} 3end {array} _ {v}

\({Q}^{\mathrm{depl}}(v)\) can be written as:

\[\]

: label: EQ-None

{int} _ {{Gamma} _ {F}}} {F}} {F} ^ {Q}cdot vdGamma

which corresponds to the particular case where \({f}^{Q}=0\) and \({u}^{Q}=0\) and where \({F}^{Q}\) is a vector that is constant on \(\gamma\) and zero elsewhere and with for each \({v}_{i}\), \({F}_{i}^{Q}=\frac{1}{\mid \gamma \mid }\); the other components of \({F}^{Q}\) being zero.

The local dual problem is written as follows:

\[\]

: label: EQ-None

{begin {array} {cccc} {sigma} {sigma} _ {mathrm {ij}, j} (omega) &text {=} & 0&mathrm {in}\ {in}\ {u} _ {i}} _ {i} (x) _ {i} (x) &text {=}, j} (omega) &text {=} & 0&mathrm {on} {gamma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {i} (x) &text {=}, & 0&mathrm {on} {gamma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} thrm {ij}} (omega) {n} _ {j} (x) &text {=} & {F} _ {i} ^ {Q} &mathrm {on} {Gamma} _ {Gamma} _ {F}end {array} &mathrm {on} {Gamma} _ {Gamma} _ {F}end {array}

Imposing \({Q}^{\mathrm{depl}}(v)\) as the loading of the dual problem is the same as imposing a linear loading in dimension 2 and a surface loading in dimension 3 restricted to \({\Gamma }_{F}\).

5.1.3. Normal displacement at one edge

Finally, the last quantity linked to the displacement is the average of the normal displacement at an edge \(\gamma \subset {\Gamma }_{F}\):

\[\]

: label: EQ-None

{Q} ^ {mathrm {depl}} (v) =frac {1} {midgammamid} {int} _ {gamma} {v} {v} _ {n} dGamma

\({Q}^{\mathrm{depl}}(v)\) can be written as:

\[\]

: label: EQ-None

{int} _ {{Gamma} _ {F}}}mathrm {pn}cdot vdGamma

where \(n\) is the normal vector at the edge \(\gamma\) and \(p=\frac{1}{\mid \gamma \mid }\) on \(\gamma\) and zero elsewhere.

The local dual problem is written as follows:

\[\]

: label: EQ-None

{begin {array} {cccc} {sigma} {sigma} _ {mathrm {ij}, j} (omega) &text {=} & 0&mathrm {in}\ {in}\ {u} _ {i}} _ {i} (x) _ {i} (x) &text {=}, j} (omega) &text {=} & 0&mathrm {on} {gamma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {i} (x) &text {=}, & 0&mathrm {on} {gamma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} {ij}} (omega) {n} _ {n} _ {j} (x) &text {=} & {mathrm {pn}} _ {i} &mathrm {on} {Gamma} _ {Gamma} _ {F}end {array}

Imposing \({Q}^{\mathrm{depl}}(v)\) as loading the dual problem is the same as imposing a loading of linear pressure in dimension 2 and surface pressure in dimension 3.

5.2. Quantities associated with constraints

5.2.1. Component of constraints in a domain

We are now interested in estimating the error on the average value of a stress component in a subdomain \(\omega\):

\[\]

: label: EQ-None

begin {array} {cc} {Q} ^ {mathrm {sigma}} (v) =frac {1} {midomegamid} {int} _ {omega} {sigma} _ {sigma} _ {mathrm {ij}} _ {mathrm {ij}} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {mathrm {ij}} _ {sigma} _ {mathrm {ij}} _ {sigma} _ {sigma} _ {mathrm {ij}

To determine \({Q}^{\mathrm{sigma}}(v)\), simply determine:

\[\]

: label: EQ-None

-a ({u} ^ {Q}, v) = {int} _ {omega}sigma (v):Sigma ({u} ^ {Q}) dOmega

which corresponds to the particular case where \({f}^{Q}=0\) and \({F}^{Q}=0\) and where \(\Sigma ({u}^{Q})\) is a symmetric operator that is constant on \(\omega\) and null elsewhere.

To do this, simply write that the constraint component is equal to the product trace between operators one of which is non-zero in \(\omega\):

\[\]

: label: EQ-None

begin {array} {cc} {Q} ^ {mathrm {ij}} ^ {mathrm {sigma}} (v) =frac {1} {omegamid} {int} _ {omega} {sigma} _ {sigma} _ {sigma} _ {omega}sigma:sigma domega +underset {sigma} _ {underset {=0} {underset} _ {underset {underbrace {}} {{int} _ {omegasetminusomega}sigma:Sigma dOmega}} = {int} _ {Omega}sigma:sigma:Sigma dOmega & i, j=Sigma dOmega & i, j=mathrm {1,2}mathrm {or} 3end {array}

with for each \({\sigma }_{\mathrm{ij}}\), \({\Sigma }_{\mathrm{ij}}=\frac{1}{\mid \omega \mid }\) for \(i=j\); \({\Sigma }_{\mathrm{ij}}=\frac{1}{2\mid \omega \mid }\) for \(i\ne j\).

The dual problem to be solved is written as follows:

\[\]

: label: EQ-None

{int} _ {omega} {sigma} _ {i}} _ {i} (v) {varepsilon} _ {mathrm {ij}} (omega) dOmega = {int} _ {omega} _ {omega} {omega} {omega} {omega} {omega}} _ {mathrm {ij}} domega

By deriving, we obtain:

\[\]

: label: EQ-None

{int} _ {omega} {(({sigma}} _ {sigma} _ {mathrm {ij}}} (omega) - {mathrm {ijkl}} {sigma} _ {mathrm {kl}}}) {mathrm {kl}}}) {v} _ {l}}}) {v} _ {i})} {v} _ {i})} _ {, j} domega - {int} _ {omega} {{omega} {{{omega}} {({sigma}}}) {v} _ {mathrm {ij}} (omega) - {a} _ {mathrm {ijkl}} {Sigma} _ {mathrm {kl}})}} _ {, j} {v} _ {v} _ {i} dOmega =0

Finally using the Stokes formula, we have:

\[\]

: label: EQ-None

{int} _ {omega} {(({sigma}} _ {sigma} _ {mathrm {ij}}} (omega) - {mathrm {ijkl}} {sigma} _ {mathrm {kl}}}) {mathrm {kl}}}) {v} _ {l}}}) {v} _ {i})} {v} _ {i})} _ {, j} domega - {int} _ {omega} ({omega}} ({sigma}} _ {l}}}) {v} _ {l}}}) {v} _ {i})} _ {i})} _ {, j} domega - {int} _ {omega} ({sigma}} _ {mathrm {ij}} (omega) - {a} _ {mathrm {ijkl}} {mathrm {ijkl}}) {n} _ {j} {v} _ {v} _ {v} _ {i} domega =0

This allows us to write the local dual problem:

\[\]

: label: EQ-None

{begin {array} {cccc} {({sigma}} _ {sigma} _ {mathrm {ij}}} (omega) - {mathrm {ijkl}}} {Sigma} _ {mathrm {kl}}})}} _ {l}})} _ {kl}})} _ {kl}})} _ {kl}})} _ {l}})} _ {, j}} &text {=} & 0&mathrm {in}Omega\ {u} _ {i} (x) &text {=} & 0&mathrm {on} {Gamma} _ {U}\ ({sigma} _ {mathrm {ij}}} (omega) - {a} _ {mathrm {ijkl}} {mathrm {ijkl}}} {mathrm {ijkl}}} (omega) - {a} _ {omega) _ {a} _ {omega) _ {mathrm {ijkl}} (x) &text {=} & 0&mathrm {on} {Gamma} _ {F}end {array}

We thus note that imposing \({Q}^{\mathrm{sigma}}(v)\) as loading the dual problem is equivalent to imposing an initial deformation \(\Sigma\). It should be noted that if we look for the error on the complete stress tensor, it will be necessary to solve as many dual problems as there are components (3 in dimension 2 and 6 in dimension 3).

5.2.2. Normal stress on one edge

It may be interesting to consider the error on the normal stress on an edge \(\gamma\):

\[\]

: label: EQ-None

begin {array} {c} {Q} ^ {mathrm {sigma}} (v) = {int} _ {gamma} ncdotsigma (v) ndomega (v) ndOmegaend {array}

This quantity does not appear in the general form. The introduction of an auxiliary function is necessary [bib25]. This \(\chi\) function is defined as \(\chi {\mid }_{\gamma }=1\) and equal to zero over the remainder of \(\partial \Omega\).

If \({n}^{\gamma }=n{\mid }_{\gamma }\), the function \(\chi\) meets the following equation:

\[\]

: label: EQ-None

begin {array} {cc} a (u,chi {n} ^ {gamma}) & = {int} _ {omega} {sigma} _ {mathrm {ij}} (u) {varepsilon}} _ {varepsilon} _ _ {mathrm {ij}} _ {gamma}} (chi {n} ^ {gamma}) d\ omega\ & = {int} _ {silon}} _ {silon} _ {silon} _ {silon} _ {gamma}} partialOmega} {sigma} _ {mathrm {ij}} (u) {n}} (u) {n} _ {j} {(chi {n} ^ {gamma})} _ {i} dOmega -underset {=0} {underset {=0}} {underset {underbrace {}} {underbrace {}} {underbrace {}} {underbrace {}} {underbrace {}} {underbrace {}} {underbrace {}} {int} _ {sigma} _ {sigma} _ {sigma} _ {mathrm {=0}} {mathrm {ij}} (u))} _ {, j} {(chi {n}} ^ {gamma})} _ {i} dOmega}}\ & = {int} _ {gamma} {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma}} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma} _ {sigma}} _ {sigma} _ {sigma}} _ {sigma} _ {sigma}} _ {sigma} _ {}

This shows that the quantity of interest can be written equivalently:

\[\]

: label: EQ-None

{Q} ^ {mathrm {sigma}} (u) = {int}} (u) = {int} _ {gamma} ncdotsigma (u) ndOmega =a (u,chi {n}} ^ {gamma}) =:tilde {Q} (u)

But as the previous equality is only true for a particular \(u\), it should be noted that:

\[\]

: label: EQ-None

{Q} ^ {mathrm {sigma}} (v) = {int}} (v) = {int} _ {gamma} ncdotsigma (v) ndOmegane a (v,chi {n}} ^ {n} ^ {gamma}) =:tilde {Q} (v)

For convenience, quantity \(\tilde{Q}(v)\) will be used instead of quantity \({Q}^{\mathrm{sigma}}(v)\). Indeed it corresponds to using \({u}^{Q}=-\chi {n}^{\gamma }\).

The quantity of interest to be used is therefore written as follows:

\[\]

: label: EQ-None

tilde {Q} (v) =-a ({u} ^ {Q}, v) = {int} _ {omega}sigma (v):Sigma (-chi {n}} ^ {gamma}) dOmega

which corresponds to the particular case where \({f}^{Q}=0\) and \({F}^{Q}=0\).

Analogous to the case of the mean of the constraints, the dual local problem is written as:

\[\]

: label: EQ-None

{begin {array} {cccc} {({sigma}} _ {sigma} _ {mathrm {ij}}} (omega) - {mathrm {ijkl}}} {Sigma} _ {mathrm {kl}}})}} _ {l}})} _ {kl}})} _ {kl}})} _ {kl}})} _ {l}})} _ {, j}} &text {=} & 0&mathrm {in}Omega\ {u} _ {i} (x) &text {=} & 0&mathrm {on} {Gamma} _ {U}\ ({sigma} _ {mathrm {ij}}} (omega) - {a} _ {mathrm {ijkl}} {mathrm {ijkl}}} {mathrm {ijkl}}} (omega) - {a} _ {omega) _ {a} _ {omega) _ {mathrm {ijkl}} (x) &text {=} & 0&mathrm {on} {Gamma} _ {F}end {array}

Imposing \(\tilde{Q}(v)\) as a load for the dual problem is like imposing an initial deformation field calculated from the displacement field \({u}^{Q}=-\chi {n}^{\gamma }\). This quantity of interest has not been implemented in*Code_Aster*.

5.2.3. Von Mises constraint in a domain

The Von Mises stress is a mechanical quantity that is very useful in terms of dimensioning for engineers. The difficulty in estimating the error on this quantity lies in the fact that its expression is not linear with respect to the displacement and therefore unusable with the technique presented. The difficulty is overcome thanks to a linearization of the expression and under certain assumptions [bib21].

The stress tensor \(\sigma\) can be broken down into the sum of a deviatoric tensor \({\sigma }^{d}\) (whose trace is zero) and a spherical tensor \({\sigma }^{s}\) (whose extra diagonal terms are zero):

\[\]

: label: EQ-None

sigma = {sigma} ^ {d} + {sigma} ^ {s}

where \({\sigma }^{d}\) and \({\sigma }^{s}\) are defined by:

\[\]

: label: EQ-None

{sigma} ^ {d} =sigma -sigma -frac {1} {3}mathrm {tr} (sigma) I {sigma} ^ {s} =frac {1} {3}mathrm {tr} (sigma) I

The Von Mises constraint, noted \({\sigma }_{\mathrm{vm}}\), is defined by:

\[\]

: label: EQ-None

{sigma} _ {mathrm {vm}} =sqrt {frac {3} {2} {sigma} ^ {d}: {sigma}: {sigma}} ^ {d}}

In linear elasticity, the stress depends linearly on the displacement, which allows us to write:

\[\]

: label: EQ-None

{sigma} ^ {d} (u) = {sigma} ^ {d} ({u} ^ {h}) + {sigma} ^ {d} (e)

Using this in the definition of the Von Mises stress, we get:

\[\]

: label: EQ-None

{sigma} _ {mathrm {vm}} (u) =sqrt {frac {3} {2}} (({sigma} ^ {d} ({u} ^ {h}) + {sigma} ^ {h}) + {sigma} ^ {h}) + {sigma} ^ {h}) + {sigma} ^ {h}) + {sigma} ^ {h}) + {sigma} ^ {h}) + {sigma} ^ {h}) + {sigma} ^ {h}) + {sigma} ^ {h}) + {sigma} ^ {h}) + {sigma} ^ {h}) + {sigma} ^ {h}) + {sigma} ^ {h}) + {sigma} ^ {h})))

The previous expression is rewritten in order to reveal a function whose limited development is known:

\[\]

: label: EQ-None

begin {array} {ccc} {sigma}} _ {mathrm {vm}} _ {mathrm {vm}}} (u) & =&sqrt {frac {3} {2} ({sigma} ^ {d} ({u}} {d}) +2 {sigma} ^ {d} ({u} ^ {d} ({u} ^ {h}) +2 {sigma} ^ {d} ({u} ^ {d} ({u} ^ {h}) +2 {sigma} ^ {d} ({u} ^ {h}) +2 {sigma} ^ {d} ({u} ^ {h}) +2 {sigma} ^ {d} ({u} ^ {h}) +2 {sigma} ^ {d} ({u}): {sigma} ^ {d} (e) + {sigma} ^ {d} (e): {sigma} ^ {d} (e))}\ & =& {sigma} _ {mathrm {vm}}} ({u} {vm}}} ({u} ^ {h}) ({u} ^ {h})sqrt {1+frac {3} {2}frac {2} {sigma} _ {mathrm {vm}}}} ({u} ^ {h}) (u} ^ {h})sqrt {1+frac {3} {2}frac {2} {sigma} _ {mathrm {vm}}}} ({u} {vm}}}} ({u} ({u} ^ {h}): {sigma} ^ {d} (e) + {sigma} ^ {d} (e): {sigma} ^ {d} (e)} {{sigma} _ {mathrm {vm}} {d}} _ {mathrm {vm}}} {vm}} {vm}}} {vm}} {vm}}} {vm}} {vm}}} {vm}}} {vm}} {vm}}} {vm}} {vm}}} {vm}} {vm}}} {vm}} {vm}}} {vm}} {vm}} {vm}}} {vm}} {vm

The expansion limited to order 1 when \(e\) tends to zero allows us to obtain the following approximation:

\[\]

: label: EQ-None

begin {array} {ccc} {sigma} _ {mathrm {vm}} _ {mathrm {vm}} _ {mathrm {vm}} ({u} ^ {h}) (1+frac {3}} {h}) (1+frac {3}} {h}) (1+frac {3}} {h}) (1+frac {3}} {2} {2}frac {sigma} ^ {h}): {sigma} ^ {h}): {sigma} ^ {h}): {sigma} ^ {h}): {sigma} ^ {h}): {sigma} ^ {h}): {sigma} ^ {h}): {sigma} ^ {h} (e)} {{sigma} _ {mathrm {vm}}} ^ {2} ({u} ^ {h})})\ &approx & {sigma} _ {mathrm {vm}}} ({u} {vm}}}} ({u} ^ {h}}}) +frac {3} {h}) +frac {3} {h})})\ &approx & {sigma} _ {mathrm {vm}}}} ({u} {vm}}}} ({u} ^ {h}}) +frac {3} {h}) +frac {3} {h}): {sigma} ^ {d} (e)} {{sigma}} {{sigma} _ {mathrm {vm}} ({u} ^ {h})}end {array}

Finally, an approximation of the error on the Von Mises constraint is determined:

\[\]

: label: EQ-None

{sigma} _ {mathrm {vm}} (u) - {sigma}} _ {mathrm {vm}} ({u} ^ {h})approxfrac {3} {2}frac {{sigma}}} (u) {sigma}} {sigma} {sigma} {2}}frac {sigma}} {sigma} {2}frac {sigma}} {sigma} {2}frac {sigma}} ^ {2}frac {sigma} ^ {h})frac {{sigma} {2}frac {sigma}} {sigma} {2}frac {sigma}} {sigma} {2}frac {sigma}} {sigma} {thrm {vm}} ({u} ^ {h})}

Thus, to estimate the error on the mean of the Von Mises stress in a domain:

\[\]

: label: EQ-None

{varepsilon} ^ {Q} =frac {1} {frac {1} {midomegamid} {int} _ {sigma} _ {mathrm {vm}} (u) - {sigma}} (u) - {sigma}} _ {sigma} _ {mathrm {vm}}} ({u} ^ {h})) dOmega

the following function can be used:

\[\]

: label: EQ-None

{Q} ^ {mathrm {vm}} (v) =frac {3} {2midomegamid} {int} _ {omega}frac {{sigma} ^ {d} ({u} {d}} ({u} ^ {h}) ({u} ^ {h}): {sigma}} ({u} {h}): {sigma}} ({u} {h}): {sigma}} ({u} {h}): {sigma}} ({u} {h}): {sigma}} ({u} {h}): {sigma}} ({u} {h}): {sigma}} ({u} {h}) ({u} {h}): {sigma}} {h})} byOmega

As for the constraint component on a domain, to determine \({Q}^{\mathrm{vm}}(v)\), you only need to determine:

\[\]

: label: EQ-None

-a ({u} ^ {Q}, v) = {int} _ {omega}Sigma ({u} ^ {Q}):sigma (v) dOmega

This corresponds to the particular case where \({f}^{Q}=0\) and \({F}^{Q}=0\) and where \(\Sigma ({u}^{Q})\) is a constant symmetric operator per element on \(\omega\) and null elsewhere. The fact that \({\sigma }^{d}(v)\) appears in the equation and not \(\sigma (v)\) requires rewriting the functional \({Q}^{\mathrm{vm}}(v)\) to make \(\sigma (v)\) appear:

\[\]

: label: EQ-None

{Q} ^ {mathrm {vm}} (v) =frac {3} {2midomegamid} {int} _ {omega}frac {{sigma} ^ {d} ({u} {d}) ({u} ^ {h}) (v) =frac {3}} {sigma}} (v))} {{sigma} ^ {d} (d)} (d) (u} ^ {d} (d)} (u}) (u} ^ {h}): (sigma (v) - {sigma} (v))} {{sigma} _ {sigma} _ {d} (d} (d)} (d) (u)} (u} ^ {d} (d)} (u)}} ({u} ^ {h})} dOmega

\[\]

: label: EQ-None

{Q} ^ {mathrm {vm}} (v) =frac {3} {2midomegamid} {int} _ {omega}frac {{sigma} ^ {d} ^ {d} ({u} {d}} ({u} ^ {d}) ({u}sigma} ^ {d}): {sigma} ^ {d}): {sigma} ^ {d} ({u}sigma} ^ {d}) ({u} ^ {d}) ({u} ^ {d}) ({u} ^ {d}) ({u} ^ {d}) ({u} ^ {d}) ({u} ^ {d}) ({u} ^ {d}) ({u} ^ {d}} ({(v)} {{sigma} _ {mathrm {vm}} ({u} ^ {h})} dOmega

Finally, the double contraction between a deviatoric tensor and a spherical tensor being zero, the functional one is written:

\[\]

: label: EQ-None

{Q} ^ {mathrm {vm}} (v) =frac {1} {midomegamid} {int} _ {omega} (frac {3} {2 {sigma} _ {sigma} _ {mathrm {vm}} _ {mathrm {vm}}} {mathrm {vm}}}} ({u} ^ {h})):sigma} ^ {h})):sigma ({u} ^ {h})):sigma ({u} ^ {h})):sigma ({u} ^ {h})):sigma (v) DOmega

with \(\Sigma ({u}^{Q})=\frac{3}{2{\sigma }_{\mathrm{vm}}({u}^{Q})}{\sigma }^{d}({u}^{Q})\) and \({u}^{Q}={u}^{h}\).

Analogous to the case of the mean of the constraints, the dual local problem is written as:

\[\]

: label: EQ-None

{begin {array} {cccc} {({sigma}} _ {mathrm {ij}}} (omega) - {a} _ {mathrm {ijkl}} {Sigma} _ {mathrm {kl}}} _ {mathrm {kl}}})}} _ {mathrm {kl}}})}} _ {mathrm {kl}}})} _ {mathrm {kl}})}} _ {mathrm {kl}})}} _ {mathrm {kl}})}} _ {mathrm {kl}})}} _ {mathrm {kl}}})} _ {l}})} _ {l}})} _ {, j}}) & 0&mathrm {on} {Gamma} _ {U} _ {U}\ ({sigma}} _ {mathrm {ij}} (omega) - {a} _ {mathrm {ijkl}}} {Sigma}} _ {Sigma}} _ {l}}) {mathrm {kl}}) {mathrm {kl}}) {mathrm {kl}}) {mathrm {kl}}) {mathrm {kl}}) {mathrm {kl}}) {mathrm {kl}}) {mathrm {kl}}) {mathrm {kl}}) {mathrm {kl}}) {mathrm {kl}}) {_ {F}end {array}

Imposing \({Q}^{\mathrm{vm}}(v)\) as a load for the dual problem is like imposing an initial deformation field.

5.3. Quantities related to fracture mechanics

5.3.1. Stress intensity factors

A method for calculating \({K}_{I}\), \({K}_{\mathrm{II}}\) and \({K}_{\mathrm{III}}\) in 2D (plane and axisymmetric) and 3D is recalled; it is based on the extrapolation of the jumps in movements on the lips of the crack. This method is only applicable to the case of flat cracks, in homogeneous and isotropic materials; it is used in command POST_K1_K2_K3.

Stress intensity factors are identified from the jump in displacements \(\left[U\right]\) by a least squares method on a segment of length \({r}_{\mathrm{max}}\). This is expressed by the following minimization problem:

\[\]

: label: EQ-None

Ktext {minimize} F (k) =frac {1} {1} {2} {2} {2} {int} _ {mathrm {max}}}} {(Cleft [U (r)right] -ksqrt {r})} ^ {2}mathrm {dr}} {left [U (r)right] -ksqrt {r})} ^ {2}mathrm {dr}

To solve this problem, a value of \(k\) is sought such that the derivative of \(F\) is cancelled out:

\[\]

: label: EQ-None

Ftext {”} (K) = {int} _ {0} _ {0}} ^ {{r} _ {mathrm {max}}}sqrt {r} (Ksqrt {r} -Cleft [U (r)right])mathrm {dr} =0

The separation of the integral gives:

\[\]

: label: EQ-None

{int} _ {0} ^ {{r} _ {mathrm {max}}} rKmathrm {dr} = {int} _ {0} ^ {{r} _ {mathrm {max}} _ {mathrm {max}}}} {mathrm {max}}}}sqrt {r}}}sqrt {r}}}sqrt {r}}}sqrt {r}}}sqrt {r}}

\[\]

: label: EQ-None

K {left [frac {{r} ^ {2}} {2}} {2}right]}} _ {0} ^ {{r} _ {mathrm {max}}}} = {int} _ {0}} ^ {0}} ^ {{r}} ^ {{r}} _ {mathrm {max}}}}sqrt {r}} _ {mathrm {max}}}}sqrt {r} Cleft [U (r)right]mathrm {dr}

The explicit formula for calculating \(K\) can be deduced directly:

\[\]

: label: EQ-None

K=frac {mathrm {2C}} {{r}} {{r} _ {mathrm {max}} ^ {2}} {int} _ {0} ^ {{r} _ {mathrm {max}}} {mathrm {max}}}}}sqrt {r}}}sqrt {r}}left [U (r)right]mathrm {r} _ {mathrm {max}}}} {mathrm {max}}}}sqrt {r}}

By carrying out the integration by a trapezius method, the previous relationship becomes:

\[\]

: label: EQ-None

K=frac {C} {{r} _ {mathrm {max}}} {mathrm {max}}} ^ {2}}sum _ {right]} _ {i-1}sqrt {{i-1}}sqrt {{r}}sqrt {{i-1}}sqrt {{r}}sqrt {{i-1}}sqrt {{r}}sqrt {{i-1}}sqrt {{r}}sqrt {{i-1}}sqrt {{r}}sqrt {{i-1}}sqrt {{r}}sqrt {{i-1}}sqrt {{r} _ {i}}) ({r} _ {i} - {r} _ {i-1})

where \(N\) is the number of nodes on segment \(\left[0,{r}_{\mathrm{max}}\right]\). Note that \(K\) is a linear form of the displacement field for a fixed \({r}_{\mathrm{max}}\), typically this corresponds to 4 or 5 elements.

To express the stress intensity factors in terms of loading for the dual problem, we will use the equation.

By showing the components of the movement on the upper lip \({U}_{i}^{\text{sup}}(r)\) and the components of the displacement on the lower lip \({U}_{i}^{\text{inf}}(r)\), we obtain:

\[\]

: label: EQ-None

{K} _ {i} =frac {2C} {{r}} {{r} _ {mathrm {max}} ^ {2}} {int} _ {gamma} ({U} _ {i} ^ {i} ^ {text {i}} ^ {text {r}} ^ {text {r}} (r))sqrt {r}mathrm {dr}

Outline \(\Gamma\) consists of the two lips of the crack; \(\Gamma\) is the upper lip and \({\Gamma }^{\text{inf}}\) is the lower lip. This makes it possible to separate the previous integral into two parts:

\[\]

: label: EQ-None

{K} _ {i} =frac {mathrm {2C}} {{r}} {{r} _ {mathrm {max}} ^ {2}}left [{int} _ {{Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {Gamma} ^ {text {inf}}} {U} _ {i} (r)sqrt {r}text {dr}right]

In order to make the test function \(v\) appear, the components of the displacements are written as the dot product of two vectors:

\[\]

: label: EQ-None

{K} _ {i} =frac {mathrm {2C}} {{r}} {{r} _ {mathrm {max}}} ^ {2}}left [{int} _ {{Gamma}} ^ {\ gamma} ^ {text {2C} ^ {text {2C}} ^ {2}}left [{int}} _ {r}}left [{int}} _ {dr} _ {gamma} ^ {gamma} ^ {gamma} ^ {text {SUP}} ^ {text {SUP}}}}} vcdot {sup}}}} vcdot {g} _ {i} (r)sqrt {r}mathrm {dr} _ {dr} {Gamma} ^ {text {inf}}}} vcdot {g} _ {i} (r)sqrt {r}mathrm {dr}right]

With \(i=I,\mathrm{II},\mathrm{III}\); \({g}_{I}\) the vector normal to the crack plane, \({g}_{\mathrm{II}}\) the vector normal to the crack background, and \({g}_{\mathrm{III}}\) the tangent vector to the crack background. This therefore means that the expression of the quantity of interest is valid in dimension 3 and therefore allows the estimation of the error of stress intensity factors for 3-dimensional problems.

Finally, by grouping all the terms together, the following expression is obtained:

\[\]

: label: EQ-None

Q (v) = {K} _ {i} (v) = {int} (v) = {int} _ {int} _ {int} _ {i} (v) = {text {sup}}cdot vmathrm {dr} (v) = {int} _ {int} _ {gamma} _ {{gamma} ^ {text {sup}}} {f} ^ {text {sup}}} {text {sup}}} {text {sup}}}cdot {inf}}}cdot vmathrm {dr} + {int} _ {int} _ {int} _ {gamma} _ {{gamma}} ^ {text {sup}}} {f} ^ {text {sup}} m {dr}

with:

\[\]

: label: EQ-None

{f} ^ {text {sup}} =frac {2C} {{2C} {{r} _ {text {max}} ^ {2}} {g} _ {i} (r)sqrt {r}

This expression corresponds to a linear loading in dimension2 (surface in dimension3) to be imposed on a partition of the lips of the crack starting from the tip and of length \({r}_{\mathrm{max}}\). The choice of ray \({r}_{\mathrm{max}}\) must be made so that it is in the singular zone and contains any Barsoum elements.