scispace - formally typeset
Search or ask a question
Journal ArticleDOI

A new hybrid explicit/implicit in-plane-out-of-plane separated representation for the solution of dynamic problems defined in plate-like domains

01 Nov 2018-Computers & Structures (Pergamon)-Vol. 210, pp 135-144
TL;DR: A new efficient hybrid explicit/implicit in-plane-out-of-plane separated representation for dynamic problems defined in plate-like domains that allows computing 3D solutions with the stability constraint exclusively determined by the coarser in-planes discretization.
About: This article is published in Computers & Structures.The article was published on 2018-11-01 and is currently open access. It has received 6 citations till now. The article focuses on the topics: Discretization & Dynamic problem.

Summary (3 min read)

1. Introduction

  • Many mechanical systems and complex structures involve plate and shell parts whose main particularity is having a characteristic dimension (the one related to the thickness) much lower that the other ones (in-plane dimensions).
  • When considering an implicit analysis, solution at each time step needs some iterations to enforce equilibrium.
  • When dynamics applies on degenerated domains, like plates or shells, and no acceptable simplifying hypotheses are available for reducing their complexity to 2D, fully 3D solutions seem compulsory.
  • Section 4 addresses time integration within the separated representation framework, and proposes an efficient hybrid explicit/implicit formulation.

2. An overview on separated representations

  • Thus, when addressing a transient model involving the unknown field uðx; tÞ, its separated representation reads [20–22] uðx; tÞ XN i¼1 XiðxÞ TiðtÞ; ð1Þ where neither the time-dependent functions TiðtÞ nor the space functions XiðxÞ are ‘‘a priori” known.
  • The spatial domain X is partially separable.
  • The complexity of the PGD simulation scales with the twodimensional mesh used to solve the BVP’s in Xxy, regardless of the mesh used in the solution of the BVP defined in Xz for calculating ZiðzÞ.
  • Domain decomposition within the separated space representation was accomplished in [25].

3.1. In-plane-out-of-plane separated representation

  • As discussed in the previous section, with X having one dimension (the one related to the thickness) much smaller than the others involving the in-plane coordinates, an in-plane-out-ofplane separated representation seems again the most appealing route for addressing 3D discretizations while keeping the computational complexity the one characteristic of 2D discretizations.
  • The separated representation constructor proceeds by computing a term of the sum at each iteration.
  • With both Un;kxy and U n;k z unknown the resulting problem becomes non-linear.
  • By introducing the trial and test functions into the weak form and then integrating in Xz because all the functions dependingon the thickness coordinate are known,weobtain a2Dweak formulation defined in Xxy whose discretization (by using a standard discretization strategy, e.g. finite elements) allows computing Un;kxy .
  • Thus, the 3D computational cost is transformed into a sequence of 2D and 1D solutions, with the associated computing time savings [5].

4. Time discretization

  • As previously commented explicit strategies are employed in many commercial codes.
  • It is important emphasizing the main aim of the present work and the proposed methodology for performing it.
  • As soon as 3D discretizations are considered, the characteristic size of the finite elements along the plate thickness becomes much smaller than the in-plane characteristic length, and then when considering explicit time integrations the time step needed for ensuring stability decreases with the throughof-thickness characteristic element length.
  • In their former works [5,6] the authors proposed in the framework of elastostatics considering in-plane-out-of-plane separated representations that allowed reducing the computational complexity of solving a fully 3D problem to the one characteristic of 2D solutions.
  • Thus, in this paper the authors analyze the intermediate procedure, the one in which the fine through-of-thickness representation is alleviated thanks to the use of the in-plane-out-of-plane space separated representation and its associated implicit unconditionally stable time integration.

4.1. Explicit-in-plane/implicit-out-of-plane hybrid scheme

  • As just indicated, in order to circumvent the just referred stability issues, the authors propose an out-of-plane implicit discretization (unconditionally stable) while the in-plane discretization (implying coarser meshes) makes use of an explicit schema.
  • It can be noticed that the derivatives involving the out-of-plane coordinate are treated using an implicit schema whereas an explicit one is retained for the in-plane derivatives.
  • Thus, the hybrid schema is some place in between standard implicit and explicit techniques, taking profit of the advantages of both them.
  • As usual in their previous works an alternated directions fixed point strategy is considered that by assuming Tkþ1 known calculates Pkþ1 and from the last updates Tkþ1.
  • When assuming Tkþ1 known the test displacement reads u ðx; y; zÞ ¼ p uðx; yÞ tkþ1u ðzÞ p vðx; yÞ tkþ1v ðzÞ p wðx; yÞ tkþ1w ðzÞ 0 B@ 1 CA ¼ P Tkþ1; ð30Þ that introduced into the weak form (29) results in a 2D problem involving the in-plane coordinates that allows calculating Pkþ1.

5.1. Dynamics of an homogeneous plate

  • In the first case study, the material occupying X is assumed homogeneous.
  • The material properties are defined in Table 1, where E is the Young modulus, m the Poisson coefficient and q the material density.
  • It is important to note that when considering fully explicit schemes, the stability is found being prescribed by the mesh size related to the thickness direction, however, when considering the hybrid schema the stability becomes given by the in-plane characteristic mesh size, that being much larger that the one related to the thickness, integration becomes more efficient.
  • To validate the hybrid approach (only in what concerns accuracy and stability, because issues related to computing time savings were addressed in [5]), the computed solution is compared with both explicit and implicit 3D finite elements integration with a time step (in the explicit case) guaranteeing the integration stability.

5.2. Considering richer out-of-plane approximations

  • In order to check the ability of the proposed technique for addressing richer out-of-plane representations, the authors consider that the domain depicted in Fig. 1 consists now in a laminated composed of 8 anisotropic plies ½0;45; 45;90 S.
  • The applied force now writes again FðtÞ ¼ 0;0;A sinðxtÞð.
  • The subscripts indicate respectively the proprieties along the longitudinal direction of the fibers (1), the in-plane transverse direction (2) and the out-of-plane direction (3).
  • The authors compared the solution obtained using the hybrid strategy with the one obtained using implicit finite elements.
  • Fig. 7 compares the time evolution of the vertical displacement at the middle of segment FG.

6. Analysis of computational performances

  • In order to investigate the performances of the proposed technique the authors perform in this section different analyses.
  • Before, the authors would like advertising on two facts.
  • 1 m and with the material properties defined in Table 1, considering the same boundary conditions than in Section 5.1 and the same loading, the last illustrated in Fig, also known as Hz ¼ 0.
  • For each mesh the authors compare the computing time employed by both the hybrid and the fully implicit PGD discretizations to solve the problem in the time interval ½0;400Dt , with the time-step Dt ¼ 10 53 s for all the simulations.
  • Here not only stability issues are addressed but also the accuracy of the computed solutions.

7. Conclusion

  • This paper proposes a new time discretization scheme for solving 3D dynamical problems defined in degenerated domains, that is, domains in which one of its characteristic dimensions is much smaller that the other ones, as it is the case when considering plates or shells.
  • Such separated representation allows the use of extremely fine descriptions along the thickness direction.
  • In this paper the authors circumvent such a drawback by using an implicit (unconditionally stable) through-the-thickness discretization whereas a standard explicit scheme is considered for treating the in-plane operators.
  • The inclusion of progressive damage models combined with dynamical effects constitutes a work in progress, where the separated representations seems an appealing option to better represent damage effects along the laminate thickness, and where explicit time integrations are usually employed in industrial applications.

Did you find this useful? Give us your feedback

Citations
More filters
Journal ArticleDOI
TL;DR: In this paper, the Proper Generalized Decomposition (PGD) technique is adopted to solve the 3D elasticity problems in a high-dimensional parametric space, which is an a priori model order reduction technique that reduces the solution of 3D partial differential equations into a set of 1D ordinary differential equations.
Abstract: The use of mesh-based numerical methods for a 3D elasticity solution of thick plates involves high computational costs. This particularly limits parametric studies and material distribution design problems because they need a large number of independent simulations to evaluate the effects of material distribution and optimization. In this context, in the current work, the Proper Generalized Decomposition (PGD) technique is adopted to overcome this difficulty and solve the 3D elasticity problems in a high-dimensional parametric space. PGD is an a priori model order reduction technique that reduces the solution of 3D partial differential equations into a set of 1D ordinary differential equations, which can be solved easily. Moreover, PGD makes it possible to perform parametric solutions in a unified and efficient manner. In the present work, some examples of a parametric elasticity solution and material distribution design of multi-directional FGM composite thick plates are presented after some validation case studies to show the applicability of PGD in such problems.

9 citations

Journal ArticleDOI
TL;DR: An enrichment procedure able to address 3D local behaviors, preserving the direct minimally-invasive coupling with existing plate and shell discretizations is proposed and will be extended to inelastic behaviors and structural dynamics.
Abstract: Most of mechanical systems and complex structures exhibit plate and shell components. Therefore, 2D simulation, based on plate and shell theory, appears as an appealing choice in structural analysis as it allows reducing the computational complexity. Nevertheless, this 2D framework fails for capturing rich physics compromising the usual hypotheses considered when deriving standard plate and shell theories. To circumvent, or at least alleviate this issue, authors proposed in their former works an in-plane-out-of-plane separated representation able to capture rich 3D behaviors while keeping the computational complexity of 2D simulations. However, that procedure it was revealed to be too intrusive for being introduced into existing commercial softwares. Moreover, experience indicated that such enriched descriptions are only compulsory locally, in some regions or structure components. In the present paper we propose an enrichment procedure able to address 3D local behaviors, preserving the direct minimally-invasive coupling with existing plate and shell discretizations. The proposed strategy will be extended to inelastic behaviors and structural dynamics.

6 citations

Journal ArticleDOI
TL;DR: This work proposes a space separation with a time adaptive number of modes to efficiently capture transient wave propagation in separable domains and shows that the PGD solution approximates its standard finite element solution counterpart with acceptable accuracy, while reducing the storage needs and the computation time.

6 citations

Journal ArticleDOI
TL;DR: This paper proposes an efficient integration of fully 3D descriptions into existing plate software to capture rich 3D behaviors while keeping the computational complexity the one of 2D simulations.
Abstract: Most of mechanical systems and complex structures exhibit plate and shell components. Therefore, 2D simulation, based on plate and shell theory, appears as an appealing choice in structural analysis as it allows reducing the computational complexity. Nevertheless, this 2D framework fails for capturing rich physics compromising the usual hypotheses considered when deriving standard plate and shell theories. To circumvent, or at least alleviate this issue, authors proposed in their former works an in-plane–out-of-plane separated representation able to capture rich 3D behaviors while keeping the computational complexity the one of 2D simulations. In the present paper we propose an efficient integration of fully 3D descriptions into existing plate software.

3 citations

Journal ArticleDOI
12 Apr 2021
TL;DR: The ESI Group’s aim is to provide real-time information about the physical properties of the Saarinen Tower and its surroundings to help engineers and scientists better understand the structure and purpose of the building.
Abstract: The need of solving industrial problems using faster and less computationally expensive techniques is becoming a requirement to cope with the present digital transformation of most industries. Recently, data is conquering the domain of engineering with different purposes: (i) defining data-driven models of materials, processes, structures and systems, whose physics-based models, when they exists, remain too inaccurate; (ii) enriching the existing physics-based models within the so-called hybrid paradigm; and (iii) using advanced machine learning and artificial intelligence techniques for scales bridging (upscaling), that is, for creating models that operating at the coarse-grained scale (cheaper in what respect the computational resources) enables integrating the fine-scale richness. The present work addresses the last item, aiming at enhancing standard structural models (defined in 2D shell geometries) for accounting all the fine-scale details (3D with rich through-the-thickness behaviors). For this purpose, two main strategies will be combined: (i) the in-plane-out-of-plane proper generalized decomposition -PGD- serving to provide the fine-scale richness; and (ii) advance machine learning techniques able to learn and extract the regression relating the input parameters with those high-resolution detailed descriptions.

2 citations


Cites background from "A new hybrid explicit/implicit in-p..."

  • ...[1-5]; (ii) thermal models defined in plates and laminates [6-7]; (iii) flows of Newtonian and non-Newtonian fluids in thin flat and rough gaps [8-12]; (iv) electromagnetism in stratified composites [13]; ....

    [...]

References
More filters
Journal ArticleDOI
TL;DR: The analyzed examples prove the potentiality and efficiency of the proposed strategy, where the computational complexity was found evolving as reported in former works, proving that 3D solutions can be computed at a 2D cost.
Abstract: The solution of 3D models in degenerated geometries in which some characteristic dimensions are much lower than the other ones -e.g. beams, plates, shells,...- is a tricky issue when using standard mesh-based discretization techniques. Separated representations allow decoupling the meshes used for approximating the solution along each coordinate. Thus, in plate or shell geometries 3D solutions can be obtained from a sequence of 2D and 1D problems allowing fine and accurate representation of the solution evolution along the thickness coordinate while keeping the computational complexity characteristic of 2D simulations. In a former work this technique was considered for addressing the 3D solution of thermoelastic problems defined in plate geometries. In this work, the technique is extended for addressing the solution of 3D elastic problems defined in shell geometries. The capabilities of the proposed approach are illustrated by considering some numerical examples involving different degrees of complexity, from simple shells to composite laminates involving stiffeners. The analyzed examples prove the potentiality and efficiency of the proposed strategy, where the computational complexity was found evolving as reported in our former works, proving that 3D solutions can be computed at a 2D cost.

55 citations


"A new hybrid explicit/implicit in-p..." refers background in this paper

  • ...In order to alleviate the associate computational complexity authors proposed few years ago computing the fully 3D solution employing an in-plane-out-of-plane separated representation whose computational complexity remains the one characteristic of 2D plate or shell simulations [5,6]....

    [...]

  • ...In our former works [5,6] we proposed in the framework of elastostatics considering in-plane-out-of-plane separated representations that allowed reducing the computational complexity of solving a fully 3D problem to the one characteristic of 2D solutions....

    [...]

  • ...In-plane-out-of-plane separated representations are particularly useful for addressing the solution of problems defined in plate [5], shell [6], beams [7] or extruded domains [23]....

    [...]

  • ...In plane-out-of-plane separated representations, revisited in the next section, allows reducing the 3D solution to a sequence of 2D (in-plane) and 1D (along the thickness) problems, as proved when considering elastostatics in plate and shell domains [5,6]....

    [...]

Journal ArticleDOI
TL;DR: In this article, a proper generalized decomposition (PGD) is presented for the layer-wise modeling of heterogeneous structures in order to reduce the number of unknowns, and the displacement field is approximated as a sum of separated functions of the in-plane coordinates x, y and the transverse coordinate z.

54 citations

Journal ArticleDOI
TL;DR: In this paper, the authors analyze the limits of 2D descriptions and justify the necessity of proceeding with 3D descriptions, and employ an advanced discretization technique making use of an efficient in-plane-out-of-plane separated representation of the different fields involved in the model.
Abstract: Nowadays thermoplastic composite materials are more and more used due to their specific excellent mechanical properties and a good recyclability. However, many difficulties are encountered during their forming processes, specially in the case of thermoplastic composites –TPC–. Therefore, consolidation of thermoplastic composites is becoming one of the most active research topics in composite manufacturing. Many processes proceed by heating prepregs to melt the polymer, then apply a compression in order to remove residual porosity trapped at the layers interfaces and consolidate the material. Thus the different layers containing the molten thermoplastic resin are compressed and squeeze flow occurs. Even if some modeling has been addressed, the flow occurring in the laminate, inside the yarns and in between the yarns requires rich 3D numerical descriptions with a fine enough description of the complex kinematics taking place in the laminate thickness. In this work we analyze the limits of lubrication based descriptions, justifying the necessity of proceeding with 3D descriptions. In order to alleviate the cost that such simulations involve, we employ an advanced discretization technique making use of an efficient in-plane-out-of-plane separated representation of the different fields involved in the model. Thus very fine descriptions are possible with a computational cost characteristic of 2D descriptions, as the ones making use of the lubrication hypotheses.

53 citations

Journal ArticleDOI
TL;DR: In this article, a finite element based on the proper generalized decomposition is presented for the analysis of bi-dimensional laminated beams, which yields to an iterative process that consists of computing a product of one-dimensional functions at each iteration.

45 citations

Journal ArticleDOI
TL;DR: In this paper, the authors proposed a new approach to solve linear elastic crack problems in plates using the so-called Proper Generalized Decomposition (PGD) method, which enables to solve the crack problem in an efficient way by obtaining a single solution in which the Poisson's ratio and the plate thickness B are non-fixed parameters.

42 citations


"A new hybrid explicit/implicit in-p..." refers background in this paper

  • ...Space separated representations where enriched with discontinuous functions for representing cracks in [15], delamination in [24] and thermal contact resistances in [10]....

    [...]

Frequently Asked Questions (14)
Q1. What have the authors contributed in "A new hybrid explicit/implicit in-plane-out-of-plane separated representation for the solution of dynamic problems defined in plate-like domains" ?

In this paper the authors introduce a new efficient hybrid explicit/implicit in-plane-out-ofplane separated representation for dynamic problems defined in plate-like domains that allows computing 3D solutions with the stability constraint exclusively determined by the coarser in-plane discretization. 

In this paper the authors circumvent such a drawback by using an implicit ( unconditionally stable ) through-the-thickness discretization whereas a standard explicit scheme is considered for treating the in-plane operators. 

The main handicap of explicit simulations is that the time step must verify the stability condition, decreasing with the element size. 

The last analysis aims at taking advantage of the superior stability performances of the implicit formulation, that a priori can use larger time-steps that the ones of explicit and hybrid formulations that are only conditionally stables. 

When dynamics applies on degenerated domains, like plates or shells, and no acceptable simplifying hypotheses are available for reducing their complexity to 2D, fully 3D solutions seem compulsory. 

This paper proposes a new time discretization scheme for solving 3D dynamical problems defined in degenerated domains, that is, domains in which one of its characteristic dimensions is much smaller that the other ones, as it is the case when considering plates or shells. 

As discussed in the previous section, with X having one dimension (the one related to the thickness) much smaller than the others involving the in-plane coordinates, an in-plane-out-ofplane separated representation seems again the most appealing route for addressing 3D discretizations while keeping the computational complexity the one characteristic of 2D discretizations. 

In many structural analysis and simulation of forming processes dynamical aspects cannot be neglected and then elastic models are replaced by their elastodynamics counterparts. 

In fact the mesh employed for discretizing the out-of-plane dimension (thickness) determines the limit time-step ensuring stability, and consequently it could become quickly unaffordable when refining the out-of-plane discretization. 

In plane-out-of-plane separated representations, revisited in the next section, allows reducing the 3D solution to a sequence of 2D (in-plane) and 1D (along the thickness) problems, as proved when considering elastostatics in plate and shell domains [5,6]. 

On the contrary explicit schemes do not require iteration as the nodal accelerations are solved directly, and from which velocities and displacements are calculated by simple integration. 

This was the route employed for deriving beam, plate and shell theories in solid mechanics, that were extended later to many other physics, like flows in narrow gaps, thermal or electromagnetic problems in laminates, among many others. 

For each mesh the authors compare the computing time employed by both the hybrid and the fully implicit PGD discretizations to solve the problem in the time interval ½0;400Dt , with the time-step Dt ¼ 10 53 s for all the simulations. 

the authors perform a comparison between the three PGD formulations (explicit, hybrid and implicit) in the time interval ½0;400Dt , with Dt ¼ 10 7 s to ensure the stability of the explicit time integration.