Space–time POD based computational vademecums for parametric studies: application to thermo-mechanical problems
- Y. Lu^{1}Email authorView ORCID ID profile,
- N. Blal^{1} and
- A. Gravouil^{1}
https://doi.org/10.1186/s40323-018-0095-6
© The Author(s) 2018
Received: 8 March 2017
Accepted: 6 January 2018
Published: 13 February 2018
Abstract
Standard numerical simulations for optimization or inverse identification of welding processes remain costly and difficult due to their multi-parametric aspect and inherent complexity. The aim of this paper is to propose a non-intrusive strategy for building computational vademecums dedicated to real-time simulations of nonlinear thermo-mechanical problems. There is in essence, a set of precomputed space–time parametric solutions (snapshots), selected by an appropriate approach in the parameter space and stored in memory as quasi-optimal reduced bases (RBs) provided by the proper orthogonal decomposition method. Once the RBs are obtained, the computational vademecums can be used online and provide real-time space–time transient nonlinear thermo-mechanical solutions for any desired parameter value. The contributions of the paper consist in a space–time RBs interpolation approach with the Grassmann manifolds method, and a localized multigrid selection method that allows an automatic selection of snapshots in the parameter areas of interest for a given level of accuracy. As application, the welding simulation is considered with a transient non-linear thermo-mechanical model using the finite element method. It is shown that the moving frame allows an optimal design of the RBs. A good efficiency of the proposed approach is demonstrated. Computational vademecums can be used for optimization or inverse identification problems of welding.
Keywords
Introduction
Despite the increasing computer efficiency over the last decades, numerical simulations of welding processes remain prohibitive in terms of CPU time and storage, due to their multi-parametric aspect. To estimate the welding quantities of interest (residual stress, distortion, etc.) depending on different input parameters (typically, the geometry, materials properties, boundary conditions and the imposed loading), larges computations have to be re-run. Thus, the construction of computational vademecums [1] (called also virtual charts [2, 3] or meta-model computations) with standard computations remains a cumbersome task.
The idea of computational vademecums consists in computing at the offline stage general solutions of a parametric problem, so as to make available a database of solutions. Engineers can then use these vademecums at the online phase for design, optimization or identification, etc. Although the offline computation can be costly, it is advantageous to lose time at this stage compared to that earned at online phase, especially in the case of repetitive tasks. This paper focuses on the construction of numerical vademecums and the reduction of the associated costs for a given level of accuracy.
As mentioned earlier, the construction of computational vademecums with standard computations based on full order models for welding simulations is out of reach. It seems important to make use of reduced-order model (ROM) techniques in order to develop models with a minimal number of degrees of freedom.
On the contrary, proper generalized decomposition (PGD) methods [16–19], i.e. a priori approaches, assume a priori a separated variable representation and do not require any prior precomputed snapshots to built the RBs. Indeed, the RBs are computed and updated during iteration procedures at the online phase, although they are usually not optimal as it is the case of POD approaches. The proper generalized decomposition is firstly introduced by Ladevèze [20] under the name “radial approximation” in the frame of the LArge Time INcrements (LATIN) method [21], for solving high nonlinear problems. It has been then successfully applied to different problems: visco-plastic problems [22–24], transient dynamic problems [25], contact problems [26], etc.
Efficient ROMs lie usually on a mixed approach between a posteriori and a priori approaches. They consist of a reduction step based on POD and an enrichment stage in order to improve the quality of these RBs. Interested readers can be referred to [27] for an enrichment technique originally proposed by Ryckelynck. This approach is of success applied to solve complex fluid flows [28, 29] and speeds up thermomechanical simulations [30, 31].
Recently, Chinesta successfully extended the separated representation, i.e. PGD methods, for solving parametric (and thus multi-dimensional) problems [16, 32–34], by adding extra-coordinates, related to model parameters or boundary conditions, to standard space–time solutions, and developed a series of computational vademecums methods for different problems in sciences and engineering such as thermal control of industrial furnaces [35], shape optimization [36], computational surgery [37], etc. These results are very encouraging. The PGD methods make possible efficient simulations of complex high-dimensional problems and the resulting computational vademecums that give real-time responses open numerous possibilities in the context of simulation based engineering, e.g. optimization or inverse identification which is also at the heart of the construction of computational vademecums for welding.
Nevertheless, one relies on the a posteriori POD approaches, because preliminary computed snapshots are sometimes available and it is obviously advantageous to use them for constructing optimal RBs instead of leaving behind the previous knowledge. As encountered in nonlinear problems, recomputing the tangent matrix makes inefficient the standard POD approaches. Although several approaches, e.g. the Discrete Empirical Interpolation Method (DEIM) [38], the hyper-reduction methods [11, 27] and the asymptotic numerical method that allows eliminating the recomputation of the tangent matrix [4, 39, 40], are introduced to accelerate the computations, real time requirements remain intractable. In addition, RBs lack robustness with respect to the variation of parameter value when one deals with parametric problems. In the works of [31], a hyper-reduced model coupled with an interpolation technique [41] based on Grassmann manifolds has been applied for the adaptation of the space RBs to tackle the robustness issue. However computing of the remaining parts (generally the time RBs) is still time-consuming work.
This work presents a methodology to overcome the above issues that go with the a posteriori approaches for space–time nonlinear parametric problems. The space and time RBs are both adapted by intensive use of the Grassmann manifolds interpolation. The recomputing of stiffness tangent matrix online can be then avoided. Once the snapshots are computed offline, there is no computation anymore and thus real-time parametric responses at online phase can be expected both in space and time. Furthermore, this approach is not intrusive, which can make available existing industrial software for different problems.
A challenge in the field of constructing the computational vademecums is the control of accuracy. Accurate solutions will be provided by computational vademecums when the retained snapshots are engendered with large training parameter. However, exhaustive generation of the RBs, at the expense of the expensive cost of the offline phase, is not possible. Unlike the PGD based computational vademecums that generate possible solutions by offline computations combined with a greedy algorithm [42–45], a posteriori approaches need adaptation step at the online stage in order to well control the error associated to the RBs for parametric studies. This paper proposes a methodology, based on a localized “multi-grid” approach (in the parameter space), allowing a quasi-optimal computational vademecum (quasi-optimal RB for a given level of accuracy), that provides reliable solutions.
The purpose of the strategy is to construct a space–time (4 dimensional) “computational vademecums” with controlled error for welding problems. To this end, the concerned transient nonlinear thermo-mechanical problem is presented in “Thermo-mechanical formulation” section. A brief review of model reduction techniques is proposed in “Design of reduced basis” section. The adaptation approach dedicated to parametric studies is detailed in “ROM adaptation for parametric studies” section. An academic example of welding is studied in terms of reducibility aspect in “Reducibility of welding simulations” section. In “Quasi-optimal computational vademecums with error control-application to welding simulations” section the quasi-optimal computational vademecums are built. Finally, This paper is closed by some remarks in “Conclusion” section.
Thermo-mechanical formulation
In this section, a standard Lagrange formulation for transient nonlinear thermo-elasto-plastic problems is presented. More particularly, an alternative formulation in the moving frame is considered. For the application in welding processes, a sequentially coupled thermo-mechanical analysis is performed. The temperature is assumed to be independent on the mechanical fields.
Thermal analysis
In order to improve the computational efficiency for processes with moving heat loading like welding, [46, 47] have proposed to solve thermal problems in moving frame, where the heat flux is fixed in space and the material flows through a reference configuration. Indeed, it is shown that for welding problems the quasi-state transient thermal problem can be simplified to a steady-state problem in a such configuration [47].
Mechanical analysis
Weak formulation of the thermo-elasto-plastic problem
Finite element semi-discretized formulation
This residual problem can be solved with the Newton–Raphson scheme and incorporated with the radial return mapping algorithm for plasticity flows. The transient thermal problem is solved by a first order time integrator.
State vector of the thermo-mechanical problem
The state vector corresponds to the minimal vector needed for storing the space–time history of all the thermal and mechanical fields. It is the only data vector one keeps at the end of the FE computations. The necessary memory for the storage completely depends on the size of \(\text {X}\). Therefore, the state vector should be chosen so that its size is as small as possible.
Design of reduced basis
This section presents the generation of the reduced basis (RB) using the popular POD method. Among the various techniques for obtaining a reduced basis, POD constructs a RB that is optimal in the sense that a certain approximation error concerning the snapshots is minimized. Thus, the space spanned by the basis from POD often gives an excellent low-dimensional approximation.
Snapshot proper orthogonal decomposition
Generally, k is chosen in such a way that a great compression is gained as far as the amount of the RB to store is concerned and the corresponding low rank approximation is sufficiently accurate. One adopts the following definition:
Definition 1
k-compressibility
A space–time field is said to be k-compressible if the number of the space–time modes needed to obtain the solution within a given tolerance is k.
ROM adaptation for parametric studies
Reduced bases often lack robustness with respect to parameter variations and updating of snapshots. This section tackles the actualization of the RB for parametric studies. A local interpolation strategy based on the Grassmann manifolds interpolation is presented and will be applied in our case to the adaptation of both the space and time RBs.
Grassmann manifolds interpolation
Let \(\mathbf{Y }_{0}\in {\mathbb {R}}^{p \times k}\), where \(k\le p\), denote a full-rank orthogonal matrix associated with a POD-basis. The matrix naturely belongs to the so-called compact Stiefel manifold [52, 53] \( ST (k,p)\) , which is defined as the set of all \(p\times k\) orthonormal matrices. Besides, the columns of \(\mathbf{Y }_{0}\) form a basis of the subspace \({ S }_{0}\) of dimension k in \({\mathbb {R}}^{p}\). This subspace \({S}_{0}\) belongs to the Grassmann manifold [52, 53] \( G (k,p)\) which is defined as the set of all k-dimensional subspaces of \({\mathbb {R}}^{p}\). Generally, each k-dimensional subspace \({ S }\) can be viewed as a point of \( G (k,p)\) at which there exists a tangent space of the same dimension [52, 53]. This tangent space, denoted by \( T \), is a “flat” space in which interpolation can be performed as usual [41] and can be represented by a matrix \(\varvec{\Gamma }\in {\mathbb {R}}^{p\times k}\).
Different analytic formulas for exponential and logarithmic mappings of matrix manifolds can be found in [55]. Coupled with a standard interpolation in the tangent space, exponential and logarithmic mappings allow defining a manifold-based interpolation (Fig. 1).
Space–time bases adaptation
Remark 1
Contrary to standard POD-Galerkin models, the proposed approach does not require new transient nonlinear FE computations with respect to the new parameter value. The space and time bases are both interpolated using Grassmann manifold interpolation, which signifies that the variables homogeneity must be retained throughout interpolations. It can be highlighted that this interpolation-based procedure presents high efficiency in terms of time-cost. This provides a practical framework for parametric studies with low cost time computations as it will be shown hereinafter, even for transient nonlinear problems.
Remark 2
Application of the proposed approach to a 2D/3D parametric case can be simply carried out by employing the corresponding FE basis in Eqs. (31) and (38) for interpolation.
Remark 3
Due to the nature of the manifold-based interpolation, the solution rebuilt with this approach presents a notably better accuracy compared to a space–time adaptation with standard Lagrange interpolation methods. This point will be illustrated in the next part.
Local interpolation strategy
The space–time adaptation of the RB highly depends on the generated snapshots in the parameter space. In order to pre-compute the snapshots with a small number of trial points and a high fidelity, a local controlled error strategy is developed. Considering a transient nonlinear thermo-mechanical problem, the prior known solutions, corresponding to different parameter values \({\mu }_{i=1,\text {n}}\in {\left[ \mu _{1},\mu _{\text {n}}\right] }\), are pre-computed with full FE method. The POD snapshots \(\varvec{{\varPhi }}_i, \varvec{\Sigma }_i, \mathbf V _i\) are provided by the truncated SVD method according to a k-compressibility criterion.
Remark 4
Though a global interpolation can indeed improve the accuracy of the rebuilt solution, the space and time bases are interpolated locally in view of the numerical cost increasing in the online phase and for the interpolation scheme stabilization.
Comparison with traditional interpolation methods
The second example consists in comparing the Grassmann interpolation with traditional methods that interpolate straightforwardly solutions stemming form FE computations (direct interpolation of snapshots). Given two snapshots calculated for two extremity values in the parameter space of thermal capacity \(\mu :{C_p}\in \left[ 432,\,900 \right] \left( \text {J} \cdot \text {kg}^{-1} \cdot \text {K}^{-1}\right) \) (see Fig. 3), the state variables are then interpolated for seven selected intermediate parameter values (called assessment points) using two different methods: piecewise linear interpolation and proposed manifold-based interpolation. Figure 4 depicts the standard 2-norm error at assessment points for both thermal and mechanical variables interpolated with these two methods. It shows that the proposed manifold-based method improves more or less the quality of interpolated solutions both in thermal and mechanical cases with respect to piecewise linear interpolation. This can be explained by the fact that the proposed method takes into account the evolution of solutions at different time which may improve the interpolation accuracy. In addition, the proposed approach seems more suitable for nonlinear cases. As shown in Fig. 4, for the thermal problem, the proposed method improves only slightly the precision, while remarkable improvement can be observed in mechanical cases.
Remark 5 For the stability of methods, the space and time bases are interpolated mode by mode in the above examples.
Reducibility of welding simulations
In this section, a numerical model of welding is presented. For that purpose, a 3D nonlinear thermo-mechanical model with a moving heat source is considered. The approach based on the moving frame, presented in the previous section, is applied to analyze this model. The reducibility of the problem is studied through a SVD analysis.
Welding finite element model
The work-piece with prescribed boundary conditions is shown in Fig. 5. The heat source moves along the line of symmetry. The used material properties as well as the load parameters are given in Table 1. All the material properties are assumed independent on the temperature.
Material properties and load data
Notation | Name | Values |
---|---|---|
\({C_p}\) | Specific heat capacity | \(432\ \text {J} \cdot \text {kg}^{-1} \cdot \text {K}^{-1}\) |
\(\lambda \) | Thermal conductivity | \(46\ \text {W} \cdot \text {m}^{-1} \cdot \text {K}^{-1}\) |
\(\alpha \) | Thermal expansion | \(1.2\times 10^{-5}\ \text {K}^{-1}\) |
E | Young’s modulus | \(210\times 10^9\ \text {Pa}\) |
\(\nu \) | Poisson ratio | 0.3 |
\(\sigma _y\) | Initial yield stress | \(300\times 10^{6}\ \text {Pa}\) |
H | Linear isotropic hardening parameter | \(21\times 10^9\ \text {Pa}\) |
Q | Heat flux | \(8\times 10^6\ \text {W}\cdot \text {m}^{-2}\) |
\(\text {V}\) | Velocity of loading | \(0.001\ \text {m}\cdot \text {s}^{-1}\) |
Geometry parameter of FE model
\(\mathbf L _\mathbf{x } \)(m) | \(\mathbf L _\mathbf{y }\)(m) | \(\mathbf L _\mathbf{z }\)(m) | E.T. | E.N. | N.N. | G.N. |
---|---|---|---|---|---|---|
0.3 | 0.1 | 0.02 | CUB8 (P1) | 7200 | 9317 | 8 |
Since the problem (geometry, material, loading, BCs) is x–z plane symmetric, only one half of the actual problem is modeled Fig. 6a. The mesh characteristics are presented in Table 2.
Reducibility of problem
The investigation of the problem reducibility is performed for the components of the state vector X defined in the previous section with a SVD analysis. For comparison purpose, the thermal solution (i.e. \(\theta \) or \(\tilde{\theta }\)) is solved in both the fixed and moving frames. In the sequel, the following definitions are used:
Definition 2
Relative SVD error
Definition 3
Energy SVD error estimator
where \(\hat{\mathbf{U }}\) designates a low rank approximation for the displacement field \(\mathbf U \in \mathbb {R}^{n\times m}\), \(E_k\) defines a cumulative associated energy with k modes and \(\mathscr {E}_{k}\) denotes a energy indicator with respect to the total associated energy \(E_{tot}\) which is equal to 1 when \(k=r=min(n,m)\).
Definition 4
Reducibility condition
In this analysis, a field is assumed to be reducible if the number of needed modes for its k-compressibility is less than \(20\%\) of the total number of modes r, i.e. \(\frac{k}{r}<20\%\).
Similarly, an energy error indicator can be defined for the thermal field. As shown in Fig. 8, for the transient state solutions, more than 50 modes are required to satisfy a energy ratio of \(99.99\%\) corresponding to a relative error of \(1\%\) in fixed frame, while only 3 modes are needed to capture the same energy in the moving frame, this amount can be reduced again to 1 with a steady state assumption in the moving frame. The reducibility of this thermal problem is significantly improved in the moving frame, since the thermal field is 3-compressible whereas it is 50-compressible in the fixed frame. Figure 9 illustrates the application of SVD analysis to the mechanical problem solved in the fixed frame. A truncation energy of \(99.99\%\) requires less than 30 modes for each of these mechanical state variables, which makes the reducibility condition be satisfied.
The moving load and flowing heat flux induce the non-reducibility of the thermal problem in the fixed frame. Whereas the resolution in the moving frame, which makes the moving load be fixed in reference configuration, leads to an hyper-reducible model. Furthermore, only one mode left with steady state assumption is required in our case. Contrary to the thermal problem, the mechanical problem is reducible in the fixed frame. Indeed, the mechanical field does not diffuse and is located in the domain of laser torch, while this is not the case for the temperature field that diffuses over time.
Quasi-optimal computational vademecums with error control-application to welding simulations
This section tackles the precision problem of computational vademecums. A multigrid based method is proposed to control the error produced in the construction of computational vademecums by Grassmann manifolds interpolation. Finally, the computational vademecums with error control are built based on the above welding model. In the following, the considered error is measured with respect to High Fidelity FE Models (HFM) and model discretization errors are not taken into account [44].
Reduced basis error indicator
This definition can be similarly applied to a state variable in the state vector. Note that the defined \(\mathscr {\epsilon }\) is a summation of the SVD-mode truncation and RB interpolation errors. The SVD-mode truncation error depends on the truncation order k chosen in such a way that the error is lower than \(1\%\). The interpolation error depends on the accuracy of the RB adaptation method and the location of pre-computed snapshots in the parameter space. The multigrid based method presented hereafter gives an automatic choice of the snapshots locations (i.e. choice of the parameter sampling points) in the parameter space in order to control and optimize the RB error.
Localized multigrid selection method
It is clear that an exhaustive generation of snapshots in the parameter space can ensure a reliable solution rebuilt by interpolation. However, the resulting offline time-cost is too much expensive. Herein, an efficient multigrid selection method that allows local refinements in the parameter space is presented.
For simplification purpose, only two parameters are considered: \(\mu \in D _{\mu }, \xi \in D _{\xi }\). We start by the coarse first-order grid, given by the four snapshots associated to the four corners of the parameter space \( D = D _{\mu }\times D _{\xi }\). The error indicator \(\mathscr {\epsilon }\) is then calculated at the assessment point located in the center of the subspace. The grid is refined only when the error for that point is greater than a critical value, by adding other snapshots to the second-order grid. The quality of the refined grid is assessed by the error indicator at each center of these sub-domains. The refinement is carried out until the error reaches the prescribed accuracy. The refinement algorithm is outlined in Algorithm 4. Figure 10 shows an example of localized refinement in a two dimensional parameter space. The above method can be extended to more than two parameters. Grid refinements should be specific to each one of state variables.
Illustration of parameter refinement
Grid order | Corners of the grid | Assessment points | Error (\(\varvec{\varvec{\sigma }}_\mathbf{VM}\)) (\(\mathbf{\%}\)) | Error test |
---|---|---|---|---|
1 | 432, 900 | 666 | 4.89 | 0 |
2 | 432, 666 | 549 | 1.5 | 0 |
2 | 666, 900 | 783 | 1.21 | 0 |
3 | 432, 549 | 490.5 | 1.19 | 0 |
3 | 549, 666 | 607.5 | 0.97 | 1 |
3 | 666, 783 | 724.5 | 0.93 | 1 |
3 | 783, 900 | 841.5 | 0.98 | 1 |
4 | 432, 490.5 | 461.25 | 0.80 | 1 |
4 | 490.5, 549 | 519.75 | 0.75 | 1 |
4D\(\otimes \)1D thermal computational vademecum
An application of the proposed method to construct computational vademecums is presented herein. The notation 4D\(\otimes \)1D means that solutions provided by the vademecums is 4D (in physical space: \(\varOmega \times \left[ 0\ t_m\right] \)) and the concerned parameter space is 1D. Parametric studies are performed with respect to the thermal capacity \(C_p(\text {J} \cdot \text {kg}^{-1} \cdot \text {K}^{-1})\,\in D =\left[ 432,\,900\right] \). The quantity of interest is chosen as residual stress induced by welding, represented here by von Mises stress. In order to satisfy a level of accuracy of \(1\%\), the localized multigrid refinement in the parameter space is shown in Fig. 11. Table 3 reports the corresponding errors at assessment points for each refinement order. As one can see, the final sampling points that should be taken into account for an accuracy of \(1\%\) are \( \{432,\ 490.5,\ 549,\ 666,\ 783,\ 900 \} \).
CPU time for a new solution
Phase | Proposed approach (6 snapshots, error \(< 1\%\)) | Standard FEM |
---|---|---|
Offline | \(\approx 42 \ \text {h}\) | – |
Online | \(<0.2\ \text {s}\) | \(\approx 7 \ \text {h}\) |
4D\(\otimes \)1D mechanical computational vademecum
RB errors of von Mises stress and plastic strain
Corners of the grid | Assessment points | Error (\(\varvec{\varvec{\sigma }}_\mathbf{MV }\)) (\(\mathbf{\%}\)) | Error (p) (\(\mathbf{\%}\)) |
---|---|---|---|
300, 350 | 325 | 1.82 | 1.62 |
350, 400 | 375 | 1.97 | 1.73 |
400, 450 | 425 | 1.89 | 1.65 |
450, 500 | 475 | 1.73 | 1.90 |
CPU time and memory (yield stress computational vademecum)
Phase | CPU time (5 snapshots, error \(<2\%\)) | Memory |
---|---|---|
Full calculations | \(\approx 35\) h | 10 Gb |
RB storage | – | 100 Mb |
Online | \(<0.2\) s | \(<500\) Mb |
RB errors of von Mises stress and plastic strain in the 2D parameter space
Assessment points | Error (\(\varvec{\sigma }_{VM}\)) (\(\%\)) | Error (p) (\(\%\)) |
---|---|---|
(549, 350) | 4.51 | 5.68 |
(783, 350) | 4.45 | 5.09 |
(783, 450) | 5.62 | 6.49 |
(490.5, 425) | 2.57 | 1.92 |
(607.5, 425) | 2.52 | 2.02 |
(607.5, 475) | 2.65 | 2.24 |
(490.5, 475) | 2.67 | 2.71 |
4D\(\otimes \)2D computational vademecum
Let \( D = D_{C} \times D_{\sigma } \) be a 2D parameter space with \(C_p(\text {J} \cdot \text {kg}^{-1} \cdot \text {K}^{-1})\in D_{C} =\left[ 432,\ 900\right] \) and \(\varvec{\sigma }_y\) (MPa) \(\in D_{\sigma } =[300,\ 500]\). This time, parametric studies are considered in the square 2D-domain \( D \). The snapshots selected by the proposed approach are shown in Fig. 14 for an guaranteed error smaller than \(7\%\) (see Table 7). A 4D\(\otimes \)2D computational vademecum (Fig. 15) is thus constructed. Only two refinements are needed to satisfy the error condition. Limited memory is required for storing the RB snapshots. Parametric space–time solutions can be provided by the 2D interpolation in real time (see Table 8).
Computational vademecum for real-time process control of welding
CPU time and memory (4D\(\otimes \)2D computational vademecum)
Phase | CPU time (14 snapshots, error \(<7\%\)) | Memory |
---|---|---|
Full calculations | \(\approx 98\) h | 10 Gb |
RB storage | – | 220 Mb |
Online | \(<0.5\) s | \(<500\) Mb |
Assuming that \(Q_i|_{i=1,2} \in [8\ 12] \times 10^6\ \text {W} \cdot \text {m}^{-2}\), the real-time computational vademecum is then constructed using the proposed approach in the 2D parameter space for an error less than \(5\%\). Real time solutions are obtained for any value of input power in the parameter space. As depicted in Fig. 17, one can modify the input power according to the real-time evolution of stress, here is an example obtained for \(Q_1=8.5, Q_2=11.5 \left( \times \ 10^6\ \text {W} \cdot \text {m}^{-2} \right) \). The online use of computational vademecum can help engineers make real time decisions for the input power of welding to control the quality of work-pieces.
Conclusion
Quasi-optimal space–time computational vademecum dedicated to parametric studies of welding process is constructed with a non-intrusive strategy. The reducibility of the full transient thermo-elasto-plastic model is studied by SVD analysis. It is shown that the reducibility of the transient thermal problem is significantly improved when the RBs are pre-computed in the moving frame. The proposed approach is based on a space–time Grassmann manifold interpolation of the reduced bases. This ensures high level of efficiency and accuracy for real time simulations and significantly reduces the high cost of computations for parametric studies. Furthermore, a localized multigrid selection method is presented. It leads to an appropriate selection of the sampling points in the parameter space that guarantee the accuracy of the space–time computational vademecum. Thus, the exhaustive generation of snapshots in the parameter space is avoided.
The proposed approach is applied to a 3D transient non-linear thermo-mechanical welding model with a moving heat Source. space–time computational vademecums are obtained for a given quantity of interest (residual stress), for both thermal and mechanical parametric studies. Excellent results have been obtained for the RBs accuracy, memory storage and the online phase computations (real time). Indeed, online phase CPU time is less than 0.5s and limited memory storage is required with a guaranteed acceptable error. As one of applications of the computational vademecum (parametric solutions), fast identification of problem parameters can be straightforwardly expected. In addition, real time process control can be expected as well by considering loading-related parameters.
Extension to multiparametric (high-dimensional) analysis is in progress with application to industrial software. In this case, the main challenge will be the development of a strategy sampling efficiently the parameter space for high-dimensional problems.
Declarations
Author's contributions
All authors have prepared the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors gratefully acknowledge AREVA and SAFRAN for funding of this work within the framework of the “Life extension and manufacturing processes” teaching and research Chair.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Not applicable.
Consent for publication
Not applicable.
Ethics approval and consent to participate
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Funding
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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References
- Chinesta F, Leygue A, Bordeu F, Aguado J, Cueto E, González D, Alfaro I, Ammar A, Huerta A. Pgd-based computational vademecum for efficient design, optimization and control. Arch Comput Methods Eng. 2013;20(1):31–59.MathSciNetView ArticleMATHGoogle Scholar
- Courard A, Néron D, Ladeveze P, Andolfatto P, Bergerot A. Virtual charts for shape optimization of structures. In: 2nd ECCOMAS Young investigators conference (YIC 2013). 2013.Google Scholar
- Vitse M, Néron D, Boucard PA. Virtual charts of solutions for parametrized nonlinear equations. Comput Mech. 2014;54(6):1529–39.MathSciNetView ArticleMATHGoogle Scholar
- Niroomandi S, Alfaro I, Cueto E, Chinesta F. Model order reduction for hyperelastic materials. Int J Numer Methods Eng. 2010;81(9):1180–206.MathSciNetMATHGoogle Scholar
- Hotelling H. Analysis of a complex of statistical variables into principal components. J Educ Psychol. 1933;24(6):417.View ArticleMATHGoogle Scholar
- Loeve M. Probability theory, the university series in higher mathematics. NJ: Princeton University Press; 1960.Google Scholar
- Lorenz EN. Empirical orthogonal functions and statistical weather prediction. Cambridge, USA: Massachusetts Institute of Technology; 1956.Google Scholar
- Karhunen K. Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Helsinki: Universitat Helsinki; 1947.MATHGoogle Scholar
- Loève M. Probability theory: foundations, random sequences. New York: D. Van Nostrand Company; 1955.MATHGoogle Scholar
- Fritzen F, Böhlke T. Three-dimensional finite element implementation of the nonuniform transformation field analysis. Int J Numer Methods Eng. 2010;84(7):803–29.MathSciNetView ArticleMATHGoogle Scholar
- Hernández J, Oliver J, Huespe AE, Caicedo M, Cante J. High-performance model reduction techniques in computational multiscale homogenization. Comput Methods Appl Mech Eng. 2014;276:149–89.MathSciNetView ArticleGoogle Scholar
- Holmes P, Lumley JL, Berkooz G. Turbulence, coherent structures, dynamical systems and symmetry. Cambridge: Cambridge university press; 1998.MATHGoogle Scholar
- Kerfriden P, Gosselet P, Adhikari S, Bordas SPA. Bridging proper orthogonal decomposition methods and augmented newton-krylov algorithms: an adaptive model order reduction for highly nonlinear mechanical problems. Comput Methods Appl Mech Eng. 2011;200(5):850–66.MathSciNetView ArticleMATHGoogle Scholar
- Kunisch K, Volkwein S. Galerkin proper orthogonal decomposition methods for parabolic problems. Numerische mathematik. 2001;90(1):117–48.MathSciNetView ArticleMATHGoogle Scholar
- Yvonnet J, He QC. The reduced model multiscale method (r3m) for the non-linear homogenization of hyperelastic media at finite strains. J Comput Phys. 2007;223(1):341–68.MathSciNetView ArticleMATHGoogle Scholar
- Chinesta F, Ammar A, Cueto E. Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models. Arch Comput Methods Eng. 2010;17(4):327–50.MathSciNetView ArticleMATHGoogle Scholar
- Chinesta F, Cueto E. PGD-based modeling of materials, structures and processes. Berlin: Springer; 2014.View ArticleGoogle Scholar
- Chinesta F, Ladeveze P, Cueto E. A short review on model order reduction based on proper generalized decomposition. Arch Comput Methods Eng. 2011;18(4):395–404.View ArticleGoogle Scholar
- Ladeveze P, Passieux JC, Néron D. The latin multiscale computational method and the proper generalized decomposition. Comput Methods Appl Mech Eng. 2010;199(21):1287–96.MathSciNetView ArticleMATHGoogle Scholar
- Ladevèze P. Sur une famille d’algorithmes en mécanique des structures. Comptes rendus des séances de l’Académie des sciences. Série 2, Mécanique-physique, chimie, sciences de l’univers, sciences de la terre. 1985;300(2):41–4.Google Scholar
- Ladevèze P. Nonlinear computational structural mechanics: new approaches and non-incremental methods of calculation. New York: Springer; 2012.Google Scholar
- Boisse P, Bussy P, Ladeveze P. A new approach in non-linear mechanics: the large time increment method. Int J Numer Methods Eng. 1990;29(3):647–63.View ArticleMATHGoogle Scholar
- Cognard JY, Ladevèze P. A large time increment approach for cyclic viscoplasticity. Int J Plast. 1993;9(2):141–57.View ArticleMATHGoogle Scholar
- Relun N, Néron D, Boucard P. A model reduction technique based on the pgd for elastic-viscoplastic computational analysis. Comput Mech. 2013;51(1):83–92.MathSciNetView ArticleMATHGoogle Scholar
- Boucinha L, Gravouil A, Ammar A. Space–time proper generalized decompositions for the resolution of transient elastodynamic models. Comput Methods Appl Mech Eng. 2013;255:67–88.MathSciNetView ArticleMATHGoogle Scholar
- Giacoma A, Dureisseix D, Gravouil A, Rochette M. A multiscale large time increment/fas algorithm with time-space model reduction for frictional contact problems. Int J Numer Methods Eng. 2014;97(3):207–30.MathSciNetView ArticleMATHGoogle Scholar
- Ryckelynck D. A priori hyperreduction method: an adaptive approach. J Comput Phys. 2005;202(1):346–66.MathSciNetView ArticleMATHGoogle Scholar
- Ammar A, Ryckelynck D, Chinesta F, Keunings R. On the reduction of kinetic theory models related to finitely extensible dumbbells. J Non-Newtonian Fluid Mech. 2006;134(1):136–47.View ArticleMATHGoogle Scholar
- Chinesta F, Ammar A, Falco A, Laso M. On the reduction of stochastic kinetic theory models of complex fluids. Model Simul Mater Sci Eng. 2007;15(6):639.View ArticleGoogle Scholar
- Chinesta F, Ammar A, Lemarchand F, Beauchene P, Boust F. Alleviating mesh constraints: model reduction, parallel time integration and high resolution homogenization. Comput Methods Appl Mech Eng. 2008;197(5):400–13.MathSciNetView ArticleMATHGoogle Scholar
- Zhang Y, Combescure A, Gravouil A. Efficient hyper reduced-order model (HROM) for parametric studies of the 3D thermo-elasto-plastic calculation. Finite Elem Anal Des. 2015;102:37–51.MathSciNetView ArticleGoogle Scholar
- Ammar A, Mokdad B, Chinesta F, Keunings R. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J Non-Newtonian Fluid Mech. 2006;139(3):153–76.View ArticleMATHGoogle Scholar
- Ammar A, Mokdad B, Chinesta F, Keunings R. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. Part ii: Transient simulation using space-time separated representations. J Non-Newtonian Fluid Mech. 2007;144(2):98–121.View ArticleMATHGoogle Scholar
- Pruliere E, Chinesta F, Ammar A. On the deterministic solution of multidimensional parametric models using the proper generalized decomposition. Math Comput Simul. 2010;81(4):791–810.MathSciNetView ArticleMATHGoogle Scholar
- Ghnatios C, Masson F, Huerta A, Leygue A, Cueto E, Chinesta F. Proper generalized decomposition based dynamic data-driven control of thermal processes. Comput Methods Appl Mech Eng. 2012;213:29–41.View ArticleGoogle Scholar
- Ammar A, Huerta A, Chinesta F, Cueto E, Leygue A. Parametric solutions involving geometry: a step towards efficient shape optimization. Comput Methods Appl Mech Eng. 2014;268:178–93.MathSciNetView ArticleMATHGoogle Scholar
- Niroomandi S, González D, Alfaro I, Bordeu F, Leygue A, Cueto E, Chinesta F. Real-time simulation of biological soft tissues: a pgd approach. Int J Numer Methods Biomed Eng. 2013;29(5):586–600.MathSciNetView ArticleGoogle Scholar
- Chaturantabut S, Sorensen DC. Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput. 2010;32(5):2737–64.MathSciNetView ArticleMATHGoogle Scholar
- Niroomandi S, Alfaro I, Cueto E, Chinesta F. Real-time deformable models of non-linear tissues by model reduction techniques. Comput Methods Programs Biomed. 2008;91(3):223–31.View ArticleGoogle Scholar
- Niroomandi S, Alfaro I, Cueto E, Chinesta F. Accounting for large deformations in real-time simulations of soft tissues based on reduced-order models. Comput Methods Programs Biomed. 2012;105(1):1–12.View ArticleGoogle Scholar
- Amsallem D, Farhat C. Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J. 2008;46(7):1803–13.View ArticleGoogle Scholar
- Maday Y, Patera AT, Turinici G. A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. J Sci Comput. 2002;17(1–4):437–46.MathSciNetView ArticleMATHGoogle Scholar
- Maday Y, Rønquist EM. A reduced-basis element method. J Sci Comput. 2002;17(1–4):447–59.MathSciNetView ArticleMATHGoogle Scholar
- Rozza G, Huynh DBP, Patera AT. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch Comput Methods Eng. 2008;15(3):229–75.MathSciNetView ArticleMATHGoogle Scholar
- Veroy K, Patera AT. Certified real-time solution of the parametrized steady incompressible navier-stokes equations: rigorous reduced-basis a posteriori error bounds. Int J Numer Methods Fluids. 2005;47(8–9):773–88.MathSciNetView ArticleMATHGoogle Scholar
- Balagangadhar D, Dorai G, Tortorelli D, of Illinois at Urbana-Champaign U. A displacement-based reference frame formulation for steady-state thermo-elasto-plastic material processes. Int J Solids Struct. 1999;36(16):2397–416.View ArticleMATHGoogle Scholar
- Rajadhyaksha SM, Michaleris P. Optimization of thermal processes using an eulerian formulation and application in laser surface hardening. Int J Numer Methods Eng. 2000;47(11):1807–23.View ArticleMATHGoogle Scholar
- Canales D, Leygue A, Chinesta F, González D, Cueto E, Feulvarch E, Bergheau JM, Huerta A. Vademecum-based gfem (v-gfem): optimal enrichment for transient problems. Int J Numer Methods Eng. 2016;108:971–89.MathSciNetView ArticleGoogle Scholar
- Bergheau J, Pont D, Leblond J. Three-dimensional simulation of a laser surface treatment through steady state computation in the heat source’s comoving frame. In: Mechanical effects of welding, Springer; 1992, p. 85–92.Google Scholar
- Gu M, Goldak J, Knight A, Bibby M. Modelling the evolution of microstructure in the heat-affected zone of steady state welds. Can Metall Q. 1993;32(4):351–61.View ArticleGoogle Scholar
- Eckart C, Young G. The approximation of one matrix by another of lower rank. Psychometrika. 1936;1(3):211–8.View ArticleMATHGoogle Scholar
- Absil PA, Mahony R, Sepulchre R. Riemannian geometry of grassmann manifolds with a view on algorithmic computation. Acta Applicandae Mathematica. 2004;80(2):199–220.MathSciNetView ArticleMATHGoogle Scholar
- Edelman A, Arias TA, Smith ST. The geometry of algorithms with orthogonality constraints. SIAM J Matrix Anal Appl. 1998;20(2):303–53.MathSciNetView ArticleMATHGoogle Scholar
- Boothby WM. An introduction to differentiable manifolds and Riemannian geometry. Houston: Gulf Professional Publishing; 2003.MATHGoogle Scholar
- Amsallem D. Interpolation on manifolds of cfd-based fluid and finite element-based structural reduced-order models for on-line aeroelastic predictions. Ph.D. thesis, Stanford University. 2010.Google Scholar
- Absil PA, Mahony R, Sepulchre R. Optimization algorithms on matrix manifolds. New Jersey: Princeton University Press; 2009.MATHGoogle Scholar
- Begelfor E, Werman M. Affine invariance revisited. In: null, IEEE; 2006. p. 2087–94.Google Scholar
- De Boor C, Ron A. Computational aspects of polynomial interpolation in several variables. Math Comput. 1992;58(198):705–27.MathSciNetView ArticleMATHGoogle Scholar
- Späth H. Two dimensional spline interpolation algorithms. Natick: AK Peters, Ltd.; 1995.MATHGoogle Scholar