Advanced separated spatial representations for hardly separable domains
01 Sep 2019-Computer Methods in Applied Mechanics and Engineering (North-Holland)-Vol. 354, pp 802-819
TL;DR: The proposed technique achieves a fully separated representation for layered domains with interfaces exhibiting waviness or deviating from planar surfaces, parallel to the coordinate plane, to make possible a simple separated representation, equivalent to others.
About: This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 2019-09-01 and is currently open access. It has received 10 citations till now. The article focuses on the topics: Representation (mathematics).
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TL;DR: In this article , a tensor decomposition (TD) based reduced-order model of the hierarchical deep learning neural networks (HiDeNN) was proposed to achieve high accuracy with orders of magnitude fewer degrees of freedom than FEM, and hence a high potential to achieve fast computations with high level of accuracy for large-size engineering and scientific problems.
12 citations
TL;DR: In this paper, the impact of surface and interface roughness on heat conduction and the effective viscosity of the interfacial polymer squeeze flow was investigated using a sort of numerical microscope.
Abstract: Composite manufacturing processes usually proceed from preimpregnated preforms that are consolidated by simultaneously applying heat and pressure, so as to ensure a perfect contact compulsory for making molecular diffusion possible. However, in practice, the contact is rarely perfect. This results in a rough interface where air could remain entrapped, thus affecting the effective thermal conductivity. Moreover, the interfacial melted polymer is squeezed flowing in the rough gap created by the fibers located on the prepreg surfaces. Because of the typical dimensions of a composite prepreg, with thickness orders of magnitude smaller than its other in-plane dimensions, and its surface roughness having a characteristic size orders of magnitude smaller than the prepreg thickness, high-fidelity numerical simulations for elucidating the impact of surface and interface roughness remain today, despite the impressive advances in computational availabilities, unattainable. This work aims at elucidating roughness impact on heat conduction and the effective viscosity of the interfacial polymer squeeze flow by using an advanced numerical strategy able to reach resolutions never attained until now, a sort of numerical microscope able to attain the scale of the smallest geometrical detail.
5 citations
TL;DR: In this paper, a NURBS-based geometry representation is combined with a fully separated representation for allying efficiency and generality by addressing complex geometries, and some numerical examples are considered to prove the potential of the proposed methodology.
Abstract: Space separation within the Proper Generalized Decomposition—PGD—rationale allows solving high dimensional problems as a sequence of lower dimensional ones. In our former works, different geometrical transformations were proposed for addressing complex shapes and spatially non-separable domains. Efficient implementation of separated representations needs expressing the domain as a product of characteristic functions involving the different space coordinates. In the case of complex shapes, more sophisticated geometrical transformations are needed to map the complex physical domain into a regular one where computations are performed. This paper aims at proposing a very efficient route for accomplishing such space separation. A NURBS-based geometry representation, usual in computer aided design—CAD—, is retained and combined with a fully separated representation for allying efficiency (ensured by the fully separated representations) and generality (by addressing complex geometries). Some numerical examples are considered to prove the potential of the proposed methodology.
5 citations
TL;DR: This work studies the effects of the Lebanese folkloric dance Dabke jumping and compares it to vertical jumping, and tackles the modeling and simulation of a tibiofemoral knee joint under impact using optimized inverse dynamics and the Brinkman model in the biphasic synovial joint domain.
4 citations
Cites background or methods from "Advanced separated spatial represen..."
...Interested readers can refer to [57] and its references therein....
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...Using the approach illustrated in [57], we define a mapping of the knee domain Ω = (x, y, z) with (x, y) a circle of center (0,0) and radius R = 4....
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...Recently, a physics-based mapping has been introduced in [57] for stratified-like domains, which is suitable for the application illustrated in Figure 14....
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TL;DR: A novel Model Order Reduction (MOR) technique is developed to compute high-dimensional parametric solutions for electromagnetic fields in synchronous machines, able to construct a virtual chart within a few minutes of off-line simulation.
Abstract: A novel Model Order Reduction (MOR) technique is developed to compute high-dimensional parametric solutions for electromagnetic fields in synchronous machines. Specifically, the intrusive version of the Proper Generalized Decomposition (PGD) is employed to simulate a Permanent-Magnet Synchronous Motor (PMSM). The result is a virtual chart allowing real-time evaluation of the magnetic vector potential as a function of the operation point of the motor, or even as a function of constructive parameters, such as the remanent flux in permanent magnets. Currently, these solutions are highly demanded by the industry, especially with the recent developments in the Electric Vehicle (EV). In this framework, standard discretization techniques require highly time-consuming simulations when analyzing, for instance, the noise and vibration in electric motors. The proposed approach is able to construct a virtual chart within a few minutes of off-line simulation, thanks to the use of a fully separated representation in which the solution is written from a series of functions of the space and parameters coordinates, with full space separation made possible by the use of an adapted geometrical mapping. Finally, excellent performances are reported when comparing the reduced-order model with the more standard and computationally costly Finite Element solutions.
3 citations
Cites methods from "Advanced separated spatial represen..."
...Such mappings were previously successfully employed to deal with apparently non-separable domains [9,10]....
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...More details about this technique can be found in [9,10,16]....
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References
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TL;DR: This survey provides an overview of higher-order tensor decompositions, their applications, and available software.
Abstract: This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.
9,227 citations
"Advanced separated spatial represen..." refers background in this paper
...In fact, that conclusion was derived from the fact that these geometries involved too many terms in the material property-separated representations when invoking the SVD or its high order counterpart, the so-called HOSVD [18]....
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Book•
08 Oct 2013
TL;DR: The present text is the first available book describing the Proper Generalized Decomposition (PGD), and provides a very readable and practical introduction that allows the reader to quickly grasp the main features of the method.
Abstract: Many problems in scientific computing are intractable with classical numerical techniques. These fail, for example, in the solution of high-dimensional models due to the exponential increase of the number of degrees of freedom.Recently, the authors of this book and their collaborators have developed a novel technique, called Proper Generalized Decomposition (PGD) that has proven to be a significant step forward. The PGD builds by means of a successive enrichment strategy a numerical approximation of the unknown fields in a separated form. Although first introduced and successfully demonstrated in the context of high-dimensional problems, the PGD allows for a completely new approach for addressing more standard problems in science and engineering. Indeed, many challenging problems can be efficiently cast into a multi-dimensional framework, thus opening entirely new solution strategies in the PGD framework. For instance, the material parameters and boundary conditions appearing in a particular mathematical model can be regarded as extra-coordinates of the problem in addition to the usual coordinates such as space and time. In the PGD framework, this enriched model is solved only once to yield a parametric solution that includes all particular solutions for specific values of the parameters. The PGD has now attracted the attention of a large number of research groups worldwide. The present text is the first available book describing the PGD. It provides a very readable and practical introduction that allows the reader to quickly grasp the main features of the method. Throughout the book, the PGD is applied to problems of increasing complexity, and the methodology is illustrated by means of carefully selected numerical examples. Moreover, the reader has free access to the Matlab software used to generate these examples.
330 citations
"Advanced separated spatial represen..." refers background or methods in this paper
...The interested reader can refer to [2, 7, 11, 20, 23, 25, 26] and the references therein for practical details on the computer implementation of separated representations....
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...The problem can therefore be solved in a separated representation as a sequence of problems in the r and s coordinate domains, by using the standard PGD procedure (see [7] for more details)....
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...needs too many modes, M, to converge, that is, for reaching a residual small enough [7]....
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...It is worth mentioning that, when using the approach based on the separation of the thermal conductivity, the usual separated representation constructor, see [7], that proceeds by computing rank-one updates, hardly converged, and each new mode produced only a very slight reduction of the residual....
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...When expressing the conductivity using both, an in-plane/out-of-plane separated form or a fully separated form, by invoking in both cases the SVD, the usual rank-one constructor [7] did not reach convergence after computing 100 modes (note that 15 modes were enough when using the procedure based on the mapping described in Section 2)....
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TL;DR: In this article, the synergy between the LATIN multiscale method and the Proper Generalized Decomposition (PGD) method is discussed, which is the key of its performances.
275 citations
"Advanced separated spatial represen..." refers background in this paper
...The interested reader can refer to [2, 7, 11, 20, 23, 25, 26] and the references therein for practical details on the computer implementation of separated representations....
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TL;DR: A new paradigm in the field of simulation-based engineering sciences (SBES) to face the challenges posed by current ICT technologies is addressed, by combining an off-line stage in which the general PGD solution, the vademecum, is computed, and an on-line phase in which real-time response is obtained as a result of the queries.
Abstract: In this paper we are addressing a new paradigm in the field of simulation-based engineering sciences (SBES) to face the challenges posed by current ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, some challenging problems remain today intractable. These problems, that are common to many branches of science and engineering, are of different nature. Among them, we can cite those related to high-dimensional problems, which do not admit mesh-based approaches due to the exponential increase of degrees of freedom. We developed in recent years a novel technique, called Proper Generalized Decomposition (PGD). It is based on the assumption of a separated form of the unknown field and it has demonstrated its capabilities in dealing with high-dimensional problems overcoming the strong limitations of classical approaches. But the main opportunity given by this technique is that it allows for a completely new approach for classic problems, not necessarily high dimensional. Many challenging problems can be efficiently cast into a multidimensional framework and this opens new possibilities to solve old and new problems with strategies not envisioned until now. For instance, parameters in a model can be set as additional extra-coordinates of the model. In a PGD framework, the resulting model is solved once for life, in order to obtain a general solution that includes all the solutions for every possible value of the parameters, that is, a sort of computational vademecum. Under this rationale, optimization of complex problems, uncertainty quantification, simulation-based control and real-time simulation are now at hand, even in highly complex scenarios, by combining an off-line stage in which the general PGD solution, the vademecum, is computed, and an on-line phase in which, even on deployed, handheld, platforms such as smartphones or tablets, real-time response is obtained as a result of our queries.
265 citations
"Advanced separated spatial represen..." refers methods in this paper
...) efficiently employed for multiple purposes: simulation, optimization, inverse analysis, uncertainty propagation and simulation-based control, all them under real-time constraints [6, 9]....
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TL;DR: In-plane–out-of-plane separated representation of the involved fields within the context of the Proper Generalized Decomposition allows solving the fully 3D model by keeping a 2D characteristic computational complexity, without affecting the solvability of the resulting multidimensional model.
175 citations
"Advanced separated spatial represen..." refers background in this paper
...For s ∈ [0, 1] X = x0 + νt, x = sh1(X), For s ∈ [1, 2] X = x0 + νt, x = (s− 1)hg + h1(X), For s ∈ [2, 3] X = x0 + νt, x = (s− 2)h2(X) + hg + h1(X), (44)...
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...s ∈ [0, 1] X = r, x = sh1(X) = sh1(r), s ∈ [1, 2] X = r, x = (s− 1)l + h1(X) = (s− 1)l + h1(r), s ∈ [2, 3] X = r, x = (s− 2)h2(X) + l + h1(X) = (s− 2)h2(r) + l + h1(r), (36)...
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...In-plane/out-of-plane separated representations are particularly useful for addressing the solution of problems defined in plate and shell geometries, [3] and [4] respectively, or extruded domains [21]....
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