Multilevel model reduction for uncertainty quantification in computational structural dynamics

Ezvan, Olivier; Batou, Anas; Soize, Christian; Gagliardini, Laurent

doi:10.1007/s00466-016-1348-1

Cited by 22 publications

(14 citation statements)

References 110 publications

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“…It is recalled that for the case

false[double-struckW false] = false[double-struckM false]

, the residual kinetic energy,

E_{k} false(\overset{u}{true} false) - E_{k}^{r} false(\overset{u}{true} false)

, is minimum, for all

\overset{u}{true}

R^{N}

. More precisely, for all

\overset{v}{true}

in subspace

S_{r}

, it can be shown that

E_{k} false(\overset{u}{true} false) - E_{k} false({\overset{u}{true}}^{r} false) = E_{k} false(\overset{u}{true} - {\overset{u}{true}}^{r} false) \leq E_{k} false(\overset{u}{true} - \overset{v}{true} false) .

The expressions for matrices

false[double-struckP false]

and

false[M^{r} false]

and for product

false[M^{r} false] \overset{u}{true}

are now given, for the case of weight matrix

false[double-struckW false] = false[double-struckM false]

and reduced basis [ B ] = [ T (0)]. It can be verified that the equality

false[double-struckW false] = false[double-struckM false]

implies low‐rank mass matrix

false[M^{r} false]

to be such that

false[M^{r} false] = false[double-struckM false] false[double-struckP false]

.…”

Section: Second Methods Proposed: Construction Of a Global‐displacemenmentioning

confidence: 99%

“…For all

boldu \in R^{N}

, let u r denote the orthogonal projection of u onto subspace

S_{r}

. It is given by

u^{r} = false[double-struckP false] boldu

, in which the orthogonal‐projection matrix

false[double-struckP false]

is an ( N × N ) real matrix that can be written as

false[double-struckP false] = false[B false] {((), {false[B false]}^{T} false[double-struckW false] false[B false])}^{- 1} {false[B false]}^{T} false[double-struckW false] .

On the other hand, the kinetic energy

E_{k} false(\overset{u}{true} false)

associated with any time‐dependent velocity vector

\overset{u}{true}

is given by

E_{k} false(\overset{u}{true} false) = 1 false/ 2 {\overset{u}{true}}^{T} false[double-struckM false] \overset{u}{true}

. The kinetic energy

E_{k}^{r} false(\overset{u}{true} false) = E_{k} false({\overset{u}{true}}^{r} false)

associated with the orthogonal projection

{\overset{u}{true}}^{r} = false[double-struckP false] \overset{u}{true}

\overset{u}{true}

is given by

E_{k}^{r} false(\overset{u}{true} false) = 1 false/ 2 {false({\overset{u}{true}}^{r} false)}^{T} false[double-struckM false] {\overset{u}{true}}^{r} = 1 false/ 2 \overset{}{true}

…”

Section: Second Methods Proposed: Construction Of a Global‐displacemenmentioning

confidence: 99%

“…Operator h r is an orthogonal projector onto the subspace of functions that are constant in each subdomain. Orthogonal‐projection matrix [ H r ] can be constructed by using the matrix [ B ] such that

false[B false] = ([], \begin{matrix} center \\ array S^{1} & array 0 & array ⋯ & array 0 \\ array 0 & array S^{2} & array ⋱ & array ⋮ \\ array ⋮ & array ⋱ & array ⋱ & array 0 \\ array 0 & array ⋯ & array 0 & array S^{N_{s}} \end{matrix}),

in which submatrix [ S i ] associated with subdomain Ω i is given by

false[S^{i} false] = ([], \begin{matrix} center \\ array T^{i} ({boldX}_{α_{i, 1}}) \\ array T^{i} ({boldX}_{α_{i, 2}}) \\ array ⋮ \\ array T^{i} ({boldX}_{α_{i, {scriptN}_{i}}}) \end{matrix}),

in which [ T i ] is a nodal transformation matrix of dimension (6 × r i ), with r i as the dimension of the kinematics of subdomain Ω i ( r i = 3 for a constant field in Ω i ). In Equation , quantity α i , j denotes the index of node j of subdomain Ω i ,

{boldX}_{α_{i, j}}

is its coordinate vector, and

N_{i}

is the number of nodes in subdomain Ω i .…”

Section: Second Methods Proposed: Construction Of a Global‐displacemenmentioning

confidence: 99%

“…The methodology has been applied to automotive vehicles and PWR fuel assemblies . It has also been extended to multilevel reduction . The present research is a further improvement and extension of this method.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Dominant vibration modes for broadband frequency analysis of multiscale structures with numerous local vibration modes

Ezvan

Zeng

Ghanem

et al. 2018

Numerical Meth Engineering

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In this paper, an extension of modal analysis in linear computational structural dynamics is presented to deal with complex structures characterized by the presence of numerous local vibration modes and for which the high modal density leads to high-dimension reduced-order models. These local modes consist of isolated vibrations that are dissociated from the global vibrations of the structure skeleton and that turn out to often be of negligible contribution to the global dynamics. Therefore, an automatic mode sorting procedure is proposed to extract the dominant modes that represent the global dynamics of the skeleton. Next, an alternative filtering methodology based on a modified eigenvalue problem is presented, which allows to build, a priori, a small-dimension reduced-order basis of dominant global modes. The two methods are compared to classic modal analysis using three applications, namely, a heterogeneous plate, a simplified nuclear fuel assembly, and a detailed boiling water reactor fuel assembly.

show abstract

“…It is recalled that for the case

false[double-struckW false] = false[double-struckM false]

, the residual kinetic energy,

E_{k} false(\overset{u}{true} false) - E_{k}^{r} false(\overset{u}{true} false)

, is minimum, for all

\overset{u}{true}

R^{N}

. More precisely, for all

\overset{v}{true}

in subspace

S_{r}

, it can be shown that

E_{k} false(\overset{u}{true} false) - E_{k} false({\overset{u}{true}}^{r} false) = E_{k} false(\overset{u}{true} - {\overset{u}{true}}^{r} false) \leq E_{k} false(\overset{u}{true} - \overset{v}{true} false) .

The expressions for matrices

false[double-struckP false]

and

false[M^{r} false]

and for product

false[M^{r} false] \overset{u}{true}

are now given, for the case of weight matrix

false[double-struckW false] = false[double-struckM false]

and reduced basis [ B ] = [ T (0)]. It can be verified that the equality

false[double-struckW false] = false[double-struckM false]

implies low‐rank mass matrix

false[M^{r} false]

to be such that

false[M^{r} false] = false[double-struckM false] false[double-struckP false]

.…”

Section: Second Methods Proposed: Construction Of a Global‐displacemenmentioning

confidence: 99%

“…For all

boldu \in R^{N}

, let u r denote the orthogonal projection of u onto subspace

S_{r}

. It is given by

u^{r} = false[double-struckP false] boldu

, in which the orthogonal‐projection matrix

false[double-struckP false]

is an ( N × N ) real matrix that can be written as

false[double-struckP false] = false[B false] {((), {false[B false]}^{T} false[double-struckW false] false[B false])}^{- 1} {false[B false]}^{T} false[double-struckW false] .

On the other hand, the kinetic energy

E_{k} false(\overset{u}{true} false)

associated with any time‐dependent velocity vector

\overset{u}{true}

is given by

E_{k} false(\overset{u}{true} false) = 1 false/ 2 {\overset{u}{true}}^{T} false[double-struckM false] \overset{u}{true}

. The kinetic energy

E_{k}^{r} false(\overset{u}{true} false) = E_{k} false({\overset{u}{true}}^{r} false)

associated with the orthogonal projection

{\overset{u}{true}}^{r} = false[double-struckP false] \overset{u}{true}

\overset{u}{true}

is given by

E_{k}^{r} false(\overset{u}{true} false) = 1 false/ 2 {false({\overset{u}{true}}^{r} false)}^{T} false[double-struckM false] {\overset{u}{true}}^{r} = 1 false/ 2 \overset{}{true}

…”

Section: Second Methods Proposed: Construction Of a Global‐displacemenmentioning

confidence: 99%

false[B false] = ([], \begin{matrix} center \\ array S^{1} & array 0 & array ⋯ & array 0 \\ array 0 & array S^{2} & array ⋱ & array ⋮ \\ array ⋮ & array ⋱ & array ⋱ & array 0 \\ array 0 & array ⋯ & array 0 & array S^{N_{s}} \end{matrix}),

in which submatrix [ S i ] associated with subdomain Ω i is given by

false[S^{i} false] = ([], \begin{matrix} center \\ array T^{i} ({boldX}_{α_{i, 1}}) \\ array T^{i} ({boldX}_{α_{i, 2}}) \\ array ⋮ \\ array T^{i} ({boldX}_{α_{i, {scriptN}_{i}}}) \end{matrix}),

{boldX}_{α_{i, j}}

is its coordinate vector, and

N_{i}

is the number of nodes in subdomain Ω i .…”

Section: Second Methods Proposed: Construction Of a Global‐displacemenmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dominant vibration modes for broadband frequency analysis of multiscale structures with numerous local vibration modes

Ezvan

Zeng

Ghanem

et al. 2018

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…The stochastic Helmholtz boundary value problem is also considered. We refer to [28] for Uncertainty Quantification for frequency responses in vibroacoustics, to [8,20,21] for model order reduction for random frequency responses in structural dynamics, and to [15,30] for the stochastic Helmholtz equation with uncertainty arising either in the forcing term or in the boundary data or in the shape of the scatterer.…”

Section: Introductionmentioning

confidence: 99%