Abstract
In vibration engineering, the differential equations of wave motions and heat conduction are usually accompanied by inhomogeneous boundary conditions in practice. Boundary inhomogeneity makes the dynamics essentially nonlinear, which prevents Hilbert space from being applied for modal decomposition. To deal with this difficulty, this paper does not treat boundary inhomogeneity as a “condition”, but almost converts it into a virtual source in conjunction with homogeneous boundary. This conversion counts mostly on the LaplaceGalerkin transform, a functional tool developed in previous works. We also explore boundary topology of this virtualsource conversion, and find that its strategy is to zero the environment and simultaneously create a spatially impulsive source on the homogeneous boundary, yielding almost the same solution. In onedimensional region, such a boundary source takes the form of Dirac delta function usually combined by its derivatives. In a sense, this paper catches how Nature really handles boundary conditions.
1. Introduction
Vibration engineering encounter a great quantity of longitudinal waves, transverse waves and heat conduction, such as parabolic or hyperbolic heatconduction dynamics, acoustic or thermoacoustic oscillations, structural vibrations, quantum mechanics, electromagnetic waves, and so on. Any of these dynamics is governed by a Laplacian operator or its higher orders in space, which is often spatially nonuniform. As the boundary condition, Dirichlet, vonNeumann, or Robin, thereof is homogeneous, its eigenfunctions constitute an admissible, real, orthogonal and complete basis in the Hilbert space of a bounded region. This basis provides modal decomposition of the dynamics for system identification, computational intelligence, model reduction, realtime processing and design purposes. However, boundary conditions are inhomogeneous in many occasions; for instance, the differential equation of heat conduction is always accompanied by inhomogeneous boundary conditions, since temperature is nonzero in nature. With boundary inhomogeneity, the dynamics is essentially nonlinear, which prevents Hilbert space from being directly introduced for modal decomposition.
To remedy such a situation in this paper, we realize the inhomogeneous boundary conditions as virtual sources in conjunction with homogeneous boundary conditions. In onedimensional cases, such a source is found to be a Dirac Delta distribution combined by its spatial derivatives on boundary. Therein, with the LaplaceGalerkin transform [13], both equations governing the interior and the boundary are integrated into a single 2D transferfunction of two independent variables: one is from the time and the other is from the space. Performing the inverse LaplaceGalerkin transform of the 2D transferfunction realizes back the dynamics into homogeneous boundary conditions with virtual sources, both of which yield the identical solution in the interior. With the homogeneous boundary resulting from the virtualsource realization, the SturmLiouville properties in Hilbert space are thus applicable for further analysis and synthesis in the modefrequency domain.
The conversion of boundary inhomogeneity into virtual source has ever been conceptually applied to identify thermal inertia by inputting vonNeumann source [2], and to derive the mechanical energy of thermoacoustics [3]. In these decades, vonNeumann boundary source was employed to obtain orderreduced modelling of combustion instabilities in rocket motors. With these hints on application, this paper systematically extends to Robin sources and studies boundary topology. Robin inhomogeneity is fascinating and necessary for real practice, since the Dirichlet or vonNeumann boundary can be considered as a degenerated version of Robin boundary [47]. Therein, the boundary is the complement of the union of the exterior and the interior of the domain under consideration, so boundary conditions rely on the interaction between the process and the environment. Although degenerative Robin simplifies numerical or experimental investigation, the actual Robin boundary should be identified in practice through measured data, as seen in [813] for examples. Therein, Robin boundaries are identified for the study of cancer destruction during hyperthermia treatment, of the optical path length in inhomogeneous tissue, and of axisymmetrical induction in heating processes, respectively. To be sure, the conversion of Robin inhomogeneity into virtual source can make these important kinds of identification more accurate and reliable, since the virtualsource conceptually suggests installing an active source in measurement to trig out desired data.
Conversion of boundary inhomogeneity into virtual source also helps computational intelligence, since boundary inhomogeneity is conventionally treated as “conditions” that constrain the spatialtemporal evolution. The legitimate spatialtemporal solutions, including the integral solutions [1415], finiteelement approximation [1617], and series solutions [1820], have to be developed toward matching the boundary conditions. In vibration engineering, modal solutions are particularly popular, since it reveals the spatiotemporal structure and results in model reduction in practice. In cases of timeinvariant environments, the response is usually computed by shifting the origin of spatial coordinate to the steadystate response, upon which the dynamics with homogeneous boundary condition can be solved by separationofvariable method, as in [21] for example. This method can also be extended to timevarying environments by stepwise sampling the temporal continuity, as in [2223] for examples. Compared with the virtualsource solution, this solution is numerical tedious and incapability of capturing sudden changes in the environment.
More importantly, virtualsource conversion lead to an inputoutput modelling that makes possible realtime signal processing. The differential equation in conjunction with the boundary inhomogeneity can give designers computertime solutions by taking the boundary inhomogeneity as constraining conditions, but it is unable to catch the realtime nature. That is, such a computertime version is offline plugged into computer simulation or calculation to merely obtain the solution as a function of a preset timespan. However, a realtime version is an emulator of natural evolution, wherein the state at the next instant is only dependent on the state and the boundary inhomogeneity at the present instant. It is unnecessary to know the history of the state and the boundary inhomogeneity to predict the future, since Nature has no memory. Galerkin projection of the converted dynamics with virtual source and homogeneous boundary onto a proper basis, such as those from Proper Orthogonal Decomposition (POD) [2428], generates an orderreduced statespace realization. Given a freely assigned sampling time, Euler discretization of the statespace realization becomes the realtime version, which interprets the dynamic nature into two times of matrix multiplication and one time of matrix addition within a sampling time.
Compared with “conditions” realization of boundary inhomogeneity, virtualsource realization provides the following merits in practice:
1) It transforms the interaction between two distributed dynamics adjacent to each other into feedback interconnection, such as thermalacoustic interaction in the fields of thermoacoustic engines [29] and combustion instabilities [30]. The construction of feedback makes possible the application of modern or classical control theory to help design and analysis.
2) Modal decomposition is applicable for computational intelligence and orderreduced modelling. It results in realtime version of distributed dynamics with inhomogeneous in digital signal processing (DSP), which can be directly programmed into a microcontroller for realtime estimation of state distribution and environmental changes, toward a newly sensing technology.
3) The realtime fashion above can still replace computertime versions as a numerical simulator programmed into a generous computer. For a long run, it can be employed for efficient management of computer memory.
4) Virtualsource realization makes possible the frequencydomain identification of boundary inhomogeneity.
5) As the control actuation is set on some boundary, the virtualsource realization generates an inputoutput model served for feedback synthesis of boundary control systems.
6) Even with temporally discontinuous or impulsive environments, exact solutions can be calculated offline with virtualsource realization.
7) Active sources on boundary can be installed for identification of Robin coefficients.
2. The considered class of dynamics
This paper considers the following three types of distributed dynamics with inhomogeneous Robin boundary conditions:
and:
${\alpha}_{1}\psi +{\beta}_{1}\nabla \psi \xb7\widehat{n}={f}_{0},{\alpha}_{2}k{\nabla}^{2}\psi +{\beta}_{2}\nabla \left(k{\nabla}^{2}\psi \right)\xb7\widehat{n}={f}_{1}\mathrm{}\mathrm{}\text{on}\mathrm{}\partial \mathrm{\Omega},\mathrm{}\mathrm{}\mathrm{}\left(\text{Transverse wave}\right),$
and:
$\alpha \psi +\beta \nabla \psi \xb7\widehat{n}=f\text{}\text{on}\mathrm{}\partial \mathrm{\Omega},\mathrm{}\mathrm{}\mathrm{}\left(\text{Heat conduction}\right).$
Therein the spatial functions $\rho $, $k$ are real and positive in the bounded region $\mathrm{\Omega}\subset {\mathfrak{R}}^{3}$ is a bounded region, wherein the distributed output is denoted by $\psi $; on the boundary $\partial \mathrm{\Omega}$, $(\alpha ,\beta )\ne \left(0.0\right)$, $({\alpha}_{1},{\beta}_{1})\ne \left(0.0\right)$, $({\alpha}_{2},{\beta}_{2})\ne \left(0.0\right)$, and the boundary inhomogeneity is denoted by $f$ or ${f}_{j}$’s.
A kind of these three dynamics involves a spatial Laplacian operator $\mathcal{a}$:
$\alpha \varphi +\beta \nabla \varphi \xb7\widehat{n}=0\mathrm{}\mathrm{}\text{on}\mathrm{}\partial \mathrm{\Omega},\mathrm{}\mathrm{}\mathrm{}\left(\text{Elastic stiffness}\right),$
or:
${\alpha}_{1}\varphi +{\beta}_{1}\nabla \varphi \xb7\widehat{n}=0,{\alpha}_{2}k{\nabla}^{2}\varphi +{\beta}_{2}\nabla \left(k{\nabla}^{2}\varphi \right)\xb7\widehat{n}=0\text{on}\mathrm{}\partial \mathrm{\Omega},\mathrm{}\mathrm{}\mathrm{}\left(\text{Bending stiffness}\right).$
With the innerproduct metric:
the Laplacian operators $\mathcal{a}$ in Eq. (4) and Eq. (5) belong to the SturmLiouville class $\mathcal{a}\in SL\left(\mathrm{\Omega}\right)$, that is, their eigenfunctions constitute a real, orthonormal, and complete basis of ${L}_{2}\left(\mathrm{\Omega}\right)$. A SturmLiouville operator is usually distinguished from its selfadjointness and the compactness of its inverse. With the Green’s second identity, it is easy to know that the elastic stiffness in Eq. (4) is a SturmLiouville operator [31]. As for the bending stiffness in Eq. (5), the following shows that it is belonging to SturmLiouville class.
Let two operators $\mathcal{P}$ and $\mathcal{Q}$ be defined by $\mathcal{P}={\rho}^{1}{\nabla}^{2}$ and $\mathcal{Q}=k{\nabla}^{2}$, then the bending stiffness $\mathcal{a}$ becomes their composite, i.e. $\mathcal{a}=\mathcal{P}\mathcal{Q}$. For any $\psi $, $\varphi \in D\left(\mathcal{a}\right)$:
since ${\alpha}_{1}\left(\mathcal{Q}\psi \right)+{\beta}_{1}\nabla \left(\mathcal{Q}\psi \right)\xb7\widehat{n}=0$ and ${\alpha}_{1}\varphi +{\beta}_{1}\nabla \varphi \xb7\widehat{n}=0$. Moreover:
since ${\alpha}_{2}\left(\mathcal{Q}\varphi \right)+{\beta}_{2}\nabla \left(\mathcal{Q}\varphi \right)\xb7\widehat{n}=0$; ${\alpha}_{2}\psi +{\beta}_{2}\nabla \psi \xb7\widehat{n}=0$. Observe that:
since both sides equal ${\int}_{\mathrm{\Omega}}k{\nabla}^{2}\psi {\xb7\nabla}^{2}\varphi dV$. Therefore, the bending stiffness operator $\mathcal{a}$ is selfadjoint. Moreover, the inverse of $\mathcal{a}$ is a compact operator in ${L}_{2}\left(\mathrm{\Omega}\right)$, since $\mathcal{a}$ is fourthorder differential operator. Therefore, its eigenfunctions constitute a real, orthonormal, and complete basis of ${L}_{2}\left(\mathrm{\Omega}\right)$. Moreover, it can be shown that both elastic stiffness and bending stiffness are positive definite in this work.
3. LaplaceGalerkin transform a functional tool
With respect to the eigenfunctions set ${\left\{{\varphi}_{\lambda}\right\}}_{\lambda \in \mathrm{\Lambda}}$ of a SturmLiouville operator $\mathcal{a}$, the Galerkin transform $\mathcal{G}$ from spatial functions to modal functions, $F\left(\lambda \right)=\mathcal{G}\left[f\right(x\left)\right]$, is defined by:
Completeness and orthonormality of ${\left\{{\varphi}_{\lambda}\right\}}_{\lambda \in \mathrm{\Lambda}}$ of countable cardinality jointly imply that the Galerkin transform $\mathcal{G}$ has a unique inverse ${\mathcal{G}}^{1}$, $f\left(x\right)={\mathcal{G}}^{1}\left[F\right(\lambda \left)\right]$:
Then, the LaplaceGalerkin transform $\mathcal{H}$ from spatialtemporal functions to modalcomplex functions is defined by the composite of the Galerkin transform $\mathcal{G}$ and the Laplace transform $\mathcal{L}$:
explicitly:
Accordingly, the inverse of LaplaceGalerkin transform ${\mathcal{H}}^{1}$ is the composite of the inverse of Laplace transform and that of Galerkin transform, that is:
explicitly:
Here the domain $\mathrm{\Gamma}$ is an infinite line parallel to the imaginary axis, whereon the integral in Eq. (13) is converged.
Denote the temporal derivative $\partial /\partial t$ by ${\mathcal{D}}_{t}$, and let $\mathcal{a}$ be a SturmLiouville operator, $\mathcal{a}\in SL\left(\mathrm{\Omega}\right)$. For the set of spatialtemporal functions with homogeneous boundary and initial, the LaplaceGalerkin transform $\mathcal{H}$ is of:
where $h$ is a ratio of two expressions of finite or some infinite length constructed from two independent variables, one standing for space and the other for time, allowing for the operations of addition, subtraction, multiplication, integer exponents in time, and fractionorder exponents in space. For example:
The LaplaceGalerkin transform and its inverse perform transformation between SturmLiouville dynamics in spacetime domain and 2D transferfunction in modefrequency domain [13]. As an example to explain 2D transferfunction, let us find the impulse response of the following longitudinal wave $\widehat{G}$:
$\psi \left(0,t\right)=0,\psi \left(\pi ,t\right)=0,0\le t<\infty ,$
$\psi \left(x,0\right)=0,\dot{\psi}\left(x,0\right)=0,0\le x\le \pi .$
The elastic stiffness${\partial}^{2}/\partial {x}^{2}$ is of eigenvalues $\mathrm{\Lambda}=\left\{\text{1,}\text{}\text{4,}\text{}\text{9,}\text{\u2026}\right\}$ associated with eigenfunctions ${\varphi}_{\lambda}\left(x\right)=\sqrt{2/\pi}\mathrm{s}\mathrm{i}\mathrm{n}\sqrt{\lambda}x$. Taking the LaplaceGalerkin transform on both sides of the differential equation with homogeneous boundary and initial yields:
that is, the 2D transferfunction of the dynamics $\widehat{G}$ is:
Correspondingly, the impulse response $g={\mathcal{H}}^{1}G$ is to be:
To check whether this solution is correct, let us give the dynamics $\widehat{G}$ the 2D unitpulse $u(x,t)=\delta \left(t\right){\sum}_{\lambda \in \mathrm{\Lambda}}{\varphi}_{\lambda}\left(x\right)$, where $\mathcal{H}u=1$. Integration of the differential equation from $t={0}^{}$ to $t={0}^{+}$ yields the initial condition: $\psi (x,0)=0$ and $\dot{\psi}(x,0)={\sum}_{\lambda \in \mathrm{\Lambda}}^{}{\varphi}_{\lambda}\left(x\right)$. Thereby, the impulse response $g$ is just the solution of the initialvalue problem:
$\psi \left(0,t\right)=0,\psi (\pi ,t)=0,$
$\psi \left(x,0\right)=0,\dot{\psi}\left(x,0\right)={\sum}_{\lambda \in \mathrm{\Lambda}}^{}{\varphi}_{\lambda}\left(x\right),$
which has the form solvable by the conventional separationofvariable method. It can be found that two solutions are identical.
4. Virtual conversion of conditions into sources on boundary
This section demonstrates how to converts the boundary inhomogeneity into virtual source in conjunction with homogeneous boundary. This conversion is analogous to the strategy of Laplace transform dealing with nonzero initial conditions, wherein the initial inhomogeneity is realized as a virtual source comprising Dirac Delta function and its derivatives. Consider the following explanatory example a normalized onedimensional wave dynamics with initial inhomogeneity:
Taking the Laplace transform on Eq. (23a) with the help of integration by parts yields:
Then taking the inverse Laplace transform on Eq. (24) yields:
$\psi \left(0,t\right)=0,\psi (\pi ,t)=0,$
$\psi \left(x,0\right)=0,\dot{\psi}(x,0)=0,$
where $\delta $ is the Dirac delta distribution. The solution to Eq. (25) is almost the same as the solution to Eq. (23); both solutions are identical in the interior $t>0$ that is an open set, but different on the boundary $t=0$ that is a closed set.
Now consider the longitudinal wave dynamics $\widehat{G}$ in Eq. (1), which involves the elastic stiffness $\mathcal{a}$ in Eq. (4). As shown in Section 2, the elastic stiffness $\mathcal{a}$ is a SturmLiouville operator $\mathcal{a}\in SL\left(\mathrm{\Omega}\right)$ under the innerproduct of Eq. (6). Let $\mathrm{\Phi}={\left\{{\varphi}_{\lambda}\right\}}_{\lambda \in \mathrm{\Lambda}}$ denote the eigenfunctions set of $\mathcal{a}$ corresponding to the eigenvalues set $\mathrm{\Lambda}\text{.}$ On the boundary $\partial \mathrm{\Omega}\text{,}$ firstly, substitution $\alpha {\varphi}_{\lambda}^{}+\beta \nabla {\varphi}_{\lambda}^{}\xb7\widehat{n}=0$ for Eqs. (1b)$\times ({\varphi}_{\lambda}^{})$ yields:
Secondly, substitution $\alpha \varphi +\beta \nabla {\varphi}_{\lambda}\bullet \widehat{n}=0$ for Eq. (1b)$\times \nabla {\varphi}_{\lambda}^{}\xb7\widehat{n}$ yields:
In general, the sum of Eq. (26)$\times \beta /(\alpha +\beta )$ and Eq. (27)$\times \alpha /(\alpha +\beta )$ is to be:
Moreover, based on the Green’s second identity:
With Eqs. (26)(29), performing LaplaceGalerkin transform $\mathcal{H}$on Eq. (1a) with inhomogeneous boundary conditions in Eq. (1b) yields:
where $\mathrm{\Psi}\left(\lambda ,s\right)\equiv \mathcal{H}\left[\psi \right(x,t\left)\right]$$\widehat{f}(x,s)\equiv \mathcal{L}\left[f\right(x,t\left)\right]$, and ${B}_{\lambda}$ is to be:
Therefore, in the sense of virtual source, the 2D transferfunction $G$ of the dynamics $\widehat{G}$ in Eq. (1) is:
With the 2D transferfunction of Eq. (32), the exact solution of Eq. (1) can be obtained even when the boundary inhomogeneity $f$ in Eq. (30) or Eq. (1b) is temporally impulsive or discontinuous.
Performing the inverse LaplaceGalerkin transform ${\mathcal{H}}^{1}$ on the Eq. (32) yields:
where $q(x,t)\equiv {\mathcal{H}}^{1}\left[Q\right(\mathrm{\lambda},s\left)\right]$. In Eq. (33), the boundary inhomogeneity $f$ in Eq. (1b) has been realized as the virtual source $q$ in Eq. (33a) in conjunction with homogeneous boundary of Eq. (33b). In the interior of the domain $\mathrm{\Omega}$, the response governed by Eq. (1) is identical to that governed by Eq. (33), since both have the same 2D transferfunction.
Next, consider the transverse wave dynamics $\widehat{G}$ in Eq. (2). For simple explanation, let us take its ondimensional version:
This dynamics involves the bending stiffness $\mathcal{a}$ in Eq. (5):
which is a SturmLiouville operator, $\mathcal{a}\in SL\left(\right[0,\mathcal{l}\left]\right)$, under the innerproduct of Eq. (6). Denote its eigenvalues set by $\mathrm{\Lambda}$ and eigenfunctions set by $\mathrm{\Phi}={\left\{{\varphi}_{\lambda}\right\}}_{\lambda \in \mathrm{\Lambda}}$.
With integration by parts (onedimensional Green’s second identity), we have:
$+\left({\varphi}_{\lambda}^{\text{'}}\left(k\psi \text{'}\text{'}\right){\varphi}_{\lambda}\left(k\psi \text{'}\text{'}\right)\text{'}+\psi \left(k{\varphi}_{\lambda}^{\text{'}\text{'}}\right)\text{'}\psi \text{'}\left(k{\varphi}_{\lambda}^{\text{'}\text{'}}\right)\right)\left{}_{x=0}\right..$
With Eq. (34b) and Eq. (35b), Eq. (36) can be rephrased to be:
With Eq. (37), performing the LaplaceGalerkin transform $\mathcal{H}$ on Eq. (34a) yields:
Then, performing the inverse LaplaceGalerkin transform ${\mathcal{H}}^{1}$ on Eq. (38) yields:
The dynamics in Eq. (34) has been converted into the dynamics in Eq. (39), wherein the virtual source in Eq. (39a) comprises a Delta function $\delta $ and the second derivative of Delta function ${\delta}^{\text{'}\text{'}}\left(\u27e8\varphi ,{\delta}^{\text{'}\text{'}}\u27e9={\varphi}^{\text{'}\text{'}}\left(0\right)\right)$ distributed at $x=0$ in conjunction with homogeneous boundary in Eqs. (39b) and (39c). Both dynamics are identical in $(0,\mathcal{l}]$, since they have the identical 2D transferfunction.
As for the virtualsource realization of the dynamics of heat conduction in Eq. (3), it is similar to that of the dynamics of longitudinal wave as shown above. The above shows how the wave and heatconduction dynamics with inhomogeneous boundary “conditions” can be almost converted to delta sources on boundary in conjunction with homogeneous boundary. Inputoutput modelling results therefrom and modal decomposition in Hilbert space becomes possible.
5. Investigation on boundary topology
In this section, the virtualsource solution and the separationofvariable solution of a normalized, onedimensional heat condition with oneside Dirichlet inhomogeneity are computed and then visualized in figures. It is expected to visualize between them the identical parts in the interior but the topological difference on the inhomogeneous boundary. Accordingly, consider the following parabolic dynamics:
Two cases of environments are considered: one is timeinvariant in which $f\left(t\right)=1$ (Case I), and the other is timevarying in which $f\left(t\right)=\mathrm{s}\mathrm{i}\mathrm{n}\omega t$ (Case II).
The virtualsource conversion, as shown in Section 4, is employed to solve the Case I and Case II. Based on the Green’s second identity:
With Eqs. (40b) and (40c), taking the LaplaceGalerkin transform on Eq. (40a) yields:
As $f\left(t\right)=\text{1}$ in Case I, $F\left(s\right)=\text{1}/s$. Then taking the inverse LaplaceGalerkin transform on Eq. (42) yields:
In case II, let $f\left(t\right)$ be $\mathrm{s}\mathrm{i}\mathrm{n}\omega t$, that is, $F\left(s\right)=\omega /({s}^{2}+{\omega}^{2})$. Taking the LaplaceGalerkin transform again on Eq. (40a) yields:
The separationofvariable method is improper to solve the solution of Case II, wherein the environmental temperature is timevarying. In Case I, the separationofvariable solution is to be:
Fig. 1 shows the orderreduced solutions of Case I with 500 modes being considered. At $t=\text{0.5}$ the separationofvariable solution in Eq. (45) is in juxtaposition with the virtualsource solution of Eq. (43) for comparison. It is found that both solutions are identical in the interior of the domain. However, close view on the inhomogeneous boundary $x=\text{0}$ as shown in Fig. 2 reveals that the exact solution ($N\to \mathrm{\infty}$) by the virtualsource conversion is discontinuous at $x=\text{0}$. With the inverse LaplaceGalerkin transform, Eq. (42) is almost equivalent to:
simultaneously with homogeneous boundary. Eq. (46) implies that the virtual source is actually impulsive on the homogeneous boundary. The virtualsource conversion realizes boundary inhomogeneity as delta sources combined by their derivatives.
Figs. 3 and 4 show the virtualsource solution for timevarying environment: $f\left(t\right)=\mathrm{s}\mathrm{i}\mathrm{n}t$. At $t=\text{0.5}$, two orderreduced solutions according to $N=\text{200}$ and $N=\text{1000}$, respectively, are juxtaposed with each other. The convergence on the boundary appears, which implies that the solution will reach discontinuity at $x=\text{0}$ as $N\to \mathrm{\infty}$. This verifies again that the virtualsource conversion is of the strategy to zero the environment and simultaneously give impulsive sources on the homogeneous boundary, yielding almost the same solution.
Fig. 1Topological comparison of the virtualsource solution with the separationofvariable solution under timeinvariant environment
Fig. 2Close view on the inhomogeneous boundary of Fig. 1
Fig. 3Topological convergence of the virtualsource solution under timevarying environment
Fig. 4Close view on the inhomogeneous boundary of Fig. 3
6. Conclusion
For wave and heatconduction dynamics, we can realize inhomogeneous boundary “conditions” as virtualsources in conjunction with homogeneous boundary. The strategy such a virtualsource conversion is to zero the environment and simultaneously to introduce an impulsive source onto the homogeneous boundary, yielding the identical response in the interior of operation domain. Therein the boundary impulsive takes the form of Dirac delta function combined by its derivatives. In practice, this virtualsource conversion helps modal decomposition, realtime processing, computational intelligence, and system identification of distributed dynamics.
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