2. The Laplace-Temps method

2.1. General principle

Let us consider the particular case of two sub-domains separated by an interface \(\mathrm{\Gamma }\): one noted \({\Omega }_{1}\), which is unbounded and which has linear behavior and the other, \({\Omega }_{2}\), bounded and possibly non-linear. The interaction effects between the two sub-domains can be represented on the interface by an impedance that we assume is defined in the Fourier or Laplace domain. In this framework, we solve the global problem in time in \({\Omega }_{2}\) by taking into account the effects of the \({\Omega }_{1}\) sub-domain by means of an external force applied on the border. This interaction force, which involves the impedance function in the time domain \(Z(t)\), corresponds to a convolutional integral with the unknown field \({u}_{\Gamma }(t)\), which will be noted \((Z\ast {u}_{\Gamma })(t)\).

Thus, in general, the evaluation of interaction efforts between sub-domains involves the calculation of a convolutional integral between two causal functions defined as:

(2.1)\[ (Z\ ast {u} _ {\ Gamma}) (t) = {\ int}) (t) = {\ int} _ {\ int} _ {\ Gamma} (\ tau)\ text {} d\ tau)\ text {} d\ tau,\ text {} 0\ le t\ le T\]

Let :Math: widehat {Z} (s) be the Laplace transform of:math: Z (t) `, we will assume that it is analytic in the complex semi-plane:math:mathit {Re e} (s) > {Re e} (s) > {sigma} _ {0} > and slowly growing for:math:mid stext {|}} `big:

:math:`

widehat{Z}(s)

le C({sigma }_{0})

s{

}^{mathrm{mu }}text{avec}C({sigma }_{0})text{et}mathrm{mu }in ℝ`

In this context, let \(\widehat{Z}(s)={\widehat{Z}}_{m}(s)\widehat{P}(s)\) with \(\widehat{P}(s)\) be a polynomial function in \(s\) with matrix values [1] _ Of degree \(m\ge \mu\):

\[\]

: label: eq-5

widehat {P} (s) =sum _ {p=0} ^ {p=0} ^ {m} {Lambda} _ {p} {s} {s} ^ {p}

If this polynomial decomposition is considered, the convolution kernel \(Z(t)\) defined in the sense of distributions can be expressed in the form of a differentiation operator of order \(p\) as:

(2.2)\[ Z (t) =\ left ({Z} _ {m}\ ast\ sum _ {p=0} ^ {m} {\ Lambda} _ {p}\ frac {{d} ^ {p} ^ {p}\ delta}\ delta} {d} {p}}\ right) (t)\]

Therefore, the convolution product can finally be written as:

(2.3)\[ (Z\ ast {u} _ {\ Gamma}) (t) =\ sum _ {p=0} ^ {m} ({\ int} _ {0} ^ {z} _ {m} _ {m} (\ tau) {\ tau) {\ tau) {\ Lambda} {\ Lambda} _ {p} {u} _ {\ gamma} ^ {(p)} {(p)} _ (t-\ tau) d\ tau)\]

with \({u}_{\Gamma }(t)\) a sufficiently differentiable causal function.

Using the Lubich quadrature method, the preceding equation can be expressed as a discrete convolution. In fact, if we write the discretization time step \(\Delta t\), it can be written at the time moments \(n\Delta t\) (\(0\le n\Delta t\le t\)) as follows:

(2.4)\[ (Z\ ast {u} _ {\ Gamma}) (n\ Delta t) =\ sum _ {k=1} ^ {n} ({\ Psi} _ {1} ^ {n-k+1} {u} {u} {u} _ {\ Gamma, k} +\ text {.} \ text {.} \ text {.} + {\ Psi} _ {p} ^ {n-k+1} {n-k+1} {u} _ {\ Gamma, k} ^ {(p)} +\ text {.} \ text {.} \ text {.} + {\ Psi} _ {m} ^ {n-k+1} {n-k+1} {u} _ {\ Gamma, k} ^ {(m)})\]

where coefficients \({\Psi }_{k}^{j}\) contain the contribution of matrices \({\Lambda }_{k}\).

2.2. Application to ground impedance operators

The ground impedance matrix, i.e. the space-discretized version of the impedance operator, relates a displacement vector defined on the degrees of freedom of the soil-structure interface (or in a generalized manner in the context of Aster-MISS3D chaining, on the border FEM - BEM) to the force vector defined on the same interface, noted \(\mathrm{\Gamma }\) in the following:

(2.5)\[ \ widehat {Z} (s) {\ widehat {u}}} _ {\ mathrm {\ Gamma}} (s) = {\ widehat {F}}} _ {\ mathrm {\ Gamma}} (s)\]

In general, the ground impedance matrix, which is assumed to be analytical in the \(\mathit{\Re e}(s)>{\sigma }_{0}\) half-space for reasons of causality, can be expressed in the following form:

(2.6)\[ \ widehat {Z} (s) = {\ widehat {Z}}} _ {\ mathit {sing}} (s) + {\ widehat {Z}}} _ {\ mathit {nsing}} (s)\]

where \({\widehat{Z}}_{\mathit{nsing}}(s)\) corresponds to the regular part of the impedance whose inverse Laplace transform \({Z}_{\mathit{nsing}}(t)\), which exists in the classical sense, is defined as an exponentially bounded and time-continuous function that cancels out for \(t<0\):

(2.7)\[ {Z} _ {\ mathit {nsing}} (t) =\ frac {1} {2\ mathrm {\ pi} i} {\ int} _ {{\ mathrm {\ sigma}}} _ {0} +iR} {0} +iR} {e} {e}} ^ {e} ^ {st} {st} {\ widehat {Z}}} _ {\ mathit {nsing}} (s)\ mathit {ds}} (s)\ mathit {ds}}\]

On the other hand, \({\widehat{Z}}_{\mathit{sing}}(s)\) refers to the singular part of the impedance, understood here as a slowly growing and unbounded function at high frequency. In particular, by analogy to a FEM formulation, this singular part of the impedance can be written as follows:

(2.8)\[ {\ widehat {Z}} _ {\ mathit {sing}} (s) = {M} _ {\ mathrm {\ Gamma}} {s} ^ {2} + {C} _ {\ mathrm {\ Gamma}} (s) = {M} _ {\ mathrm {\ Gamma}} _ {\ mathrm {\ Gamma}} _ {\ mathrm {\ Gamma}} _ {\ mathrm {\ Gamma}}\]

where \({M}_{\mathrm{\Gamma }}\), \({C}_{\mathrm{\Gamma }}\) and \({K}_{\mathrm{\Gamma }}\) respectively model the contributions of the ground in terms of inertia, damping, and stiffness at the \(\mathrm{\Gamma }\) interface. Two interesting points to mention are highlighted. The first aims to emphasize that this formulation makes it easy to show that the impedance explicitly verifies the condition explained in equation () for \(m=2\). The second relates rather to the non-singular nature of the inverse Laplace transform, which this time only exists in the sense of the distributions and which is therefore written using the Dirac delta and its first and second order derivatives:

(2.9)\[ {Z} _ {\ mathit {sing}} (t) = {M}} (t) = {M} _ {\ mathrm {\ Gamma}}\ ddot {\ mathrm {\ delta}} (t) + {C} _ {\ mathrm {\ Gamma}} (t) + {\ mathrm {\ delta}} (t) + {\ mathrm {\ delta}} (t) + {\ mathrm {\ delta}} (t) + {\ mathrm {\ Gamma}}\ mathrm {\ delta} (t)\]

The numerical manipulation of \({\widehat{Z}}_{\mathit{sing}}(t)\) is very delicate and at the same time essential, because this evolution must be calculated in order to obtain the transient soil-structure interaction forces. Therefore, it can be said that the particular nature of the temporal impedance makes it necessary to use more efficient time discretization approaches and among these we find the Laplace-Temps method based on a convolutional quadrature method.

In this context, the convolutional quadrature method presented by Lubich can be applied for the numerical evaluation of a convolution integral as given in equation (), where for \(Z(t)\) the Laplace transform is considered the inverse of the ground impedance \(\widehat{Z}(s)\) and \({u}_{\mathrm{\Gamma }}(t)\) the displacement field on the interface. This method makes it possible to evaluate equation () in an approximate way by a discrete convolution (with a time step \(\mathrm{\Delta }t>0\)):

(2.10)\[ (Z\ ast {u} _ {\ mathrm {\ Gamma}}) (n\ mathrm {\ Gamma}}}) (n\ mathrm {\ Delta} t\ le t} {\ Phi} _ {\ Phi} _ {k} {u} _ {\ mathrm {\ Gamma}}} (t-n\ mathrm {\ Delta} t)\]

where the coefficients \({\Phi }_{k}\) correspond to the weights of the following power series:

(2.11)\[ \ sum _ {k=0} ^ {+\ infty} {\ infty} {\ Phi} _ {\ Phi} _ {k} _ {\ mathrm {\ Delta} t} =\ widehat {Z} ({s} _ {\ mathrm {\ Delta} t})\]

Sample points \({s}_{\mathrm{\Delta }t}\) for dynamic ground impedance are given in the next section.

At this point and taking into account that the expression () is homogeneous to a force, it seems appropriate to express this convolution not only in terms of displacement, but also as a function of accelerations and speeds. The proposed approach fits into this perspective and is based on the factorization of the polynomial part \(\widehat{P}(s)\) of the impedance:

(2.12)\[ \ widehat {Z} (s) = {\ widehat {Z}}} _ {m} (s)\ widehat {P} (s) = {\ widehat {Z}} _ {m} (s) ({\ stackrel {~} {M}}} _ {\ stackrel {~} {M}}} _ {\ stackrel {~}}} _ {\ stackrel {~} {M}}} _ {\ stackrel {~} {M}}} _ {\ stackrel {~} {M}} C}} _ {\ mathrm {\ Gamma}} s+ {\ stackrel {~} {K}}} _ {\ mathrm {\ Gamma}})\]

Where \({\stackrel{~}{K}}_{\mathrm{\Gamma }}\), \({\stackrel{~}{C}}_{\mathrm{\Gamma }}\), and \({\stackrel{~}{M}}_{\mathrm{\Gamma }}\) are estimators of the stiffness, damping, and inertia matrices of the ground, respectively. Thus, convolution can be written using the inverse Laplace transform as follows:

(2.13)\[ (Z\ ast {u} _ {\ mathrm {\ sigma}}) (t) (t) =\ frac {1} {2\ mathrm {\ pi} i} {\ int} _ {\ mathrm {\ sigma}}} _ {0} +iR} _ {\ widehat {Z}}} _ {m} (s)\ widehat {P} (s) {\ mathrm {\ sigma}}}} _ {\ widehat {P} (s) {\ widehat} (s) {\ widehat} (s) {\ widehat} {u}} _ {\ mathrm {\ Gamma}} (s) {e} ^ {st}\ mathit {ds}}\]

Therefore, the polynomial function \(\widehat{P}(s)\) being seen as a differentiation operator that acts on the displacement field, equation () becomes:

(2.14)\[ (Z\ ast {u} _ {\ mathrm {\ Gamma}}) (t) = ({Z} _ {m}\ ast {\ stackrel {~} {M}} _ {\ mathrm {\ Gamma}}} {\ ddot {\ Gamma}}}) (t) + ({Z} _ {m}\ mathrm {\ Gamma}}} _ {m}\ ast {\ Gamma}}} {\ ddot {u}}} _ {m}\ ast {\ stackrel}} {~} {C}} _ {\ mathrm {\ Gamma}}} {\ dot {u}} _ {\ mathrm {\ Gamma}}) (t) + ({Z} _ {m}\ ast {\ stackrel {\ stackrel {\ stackrel {\ Stackrel {~} {K}}}} _ {\ mathrm {\ Gamma}}) (t)\]

where interaction efforts (noted below by \({R}_{\mathrm{\Gamma }}(t)\)) are calculated by convolutions with accelerations, speeds and displacements of the interface.

By taking a discretization time step \(\mathrm{\Delta }t>0\), the convolutional integral becomes a special case of equation () for \(m=2\):

(2.15)\[ {R} _ {n} = (Z\ ast {u} _ {\ Gamma}) (n\ Delta t) =\ sum _ {k=1} ^ {n} ({\ Psi} _ {2} ^ {n-k+1} ^ {n-k+1}} {\ ddot {k+1}} {\ n-k+1}} ^ {n-k+1}} ^ {n-k+1} ^ {n-k+1}} ^ {n-k+1} {\ dot {k+1}} {\ n-k+1}} {\ dot {k+1}} {\ dot {k+1}} {\ dot {k+1}} {\ dot {u} +1} {\ dot {u} +1} {\ dot {u} +1} {\ dot {u} +1} {\ dot {u} +1} {\ dot {u} Gamma, k} + {\ Psi} _ {0} _ {0} ^ {n-k+1} {u} _ {\ Gamma, k})\]

where the matrices multiplying the vectors of movement, speed, and acceleration are given by:

\[\]

: label: eq-20

{Psi} _ {0} ^ {k} = {Z} = {Z} _ {m} ^ {k} {stackrel {~} {K}}} _ {mathrm {Gamma}} _ {mathrm {Gamma}} {Psi} _ {1} ^ {k} = {Z} = {Z} _ {m} ^ {k} {stackrel {~} {C}}} _ {mathrm {Gamma}} _ {mathrm {Gamma}}

In fact, in a manner similar to the reasoning followed for equations () and (), the coefficients of the matrices \({\Psi }_{0}^{k}\), \({\Psi }_{1}^{k}\) and \({\Psi }_{2}^{k}\) correspond to the weights of the following power series:

\[\]

: label: eq-21

sum _ {k=0} ^ {+mathrm {infty}} {mathrm {infty}}} {mathrm {Psi}} ^ {k} = {widehat {Z}}} = {widehat {Z}}}} _ {m}} _ {m} ({m}} ({s}) _ {s} _ {Delta} t}) {mathrm {Lambda}} ^ {k} = {widehat {Z}}} {Z}} ^ {k} = {widehat {Z}}}} _ {m}} _ {m} ({m}} ({s}} _ {s} _ {Delta} t}) {mathrm

With \({\mathrm{\Lambda }}_{0}={\stackrel{~}{K}}_{\mathrm{\Gamma }}\), \({\mathrm{\Lambda }}_{1}={\stackrel{~}{C}}_{\mathrm{\Gamma }}\), and \({\mathrm{\Lambda }}_{2}={\stackrel{~}{M}}_{\mathrm{\Gamma }}\).

[1]

To simplify the theoretical approach, we consider in the following all the quantities already discretized in space (for example, with a finite element method).