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Probabilistic Deep Learning for Real-Time Large Deformation Simulations Deshpande, Saurabh ; Lengiewicz, Jakub ; Bordas, Stéphane in Computer Methods in Applied Mechanics and Engineering (2022), 398(0045-7825), 115307 For many novel applications, such as patient-specific computer-aided surgery, conventional solution techniques of the underlying nonlinear problems are usually computationally too expensive and are ... [more ▼] For many novel applications, such as patient-specific computer-aided surgery, conventional solution techniques of the underlying nonlinear problems are usually computationally too expensive and are lacking information about how certain can we be about their predictions. In the present work, we propose a highly efficient deep-learning surrogate framework that is able to accurately predict the response of bodies undergoing large deformations in real-time. The surrogate model has a convolutional neural network architecture, called U-Net, which is trained with force–displacement data obtained with the finite element method. We propose deterministic and probabilistic versions of the framework. The probabilistic framework utilizes the Variational Bayes Inference approach and is able to capture all the uncertainties present in the data as well as in the deep-learning model. Based on several benchmark examples, we show the predictive capabilities of the framework and discuss its possible limitations. [less ▲] Detailed reference viewed: 93 (5 UL)Efficient optimization-based quadrature for variational discretization of nonlocal problems ; Shen, Zhaoxiang ; et al in Computer Methods in Applied Mechanics and Engineering (2022), 396 Casting nonlocal problems in variational form and discretizing them with the finite element (FE) method facilitates the use of nonlocal vector calculus to prove well-posedness, convergence, and stability ... [more ▼] Casting nonlocal problems in variational form and discretizing them with the finite element (FE) method facilitates the use of nonlocal vector calculus to prove well-posedness, convergence, and stability of such schemes. Employing an FE method also facilitates meshing of complicated domain geometries and coupling with FE methods for local problems. However, nonlocal weak problems involve the computation of a double-integral, which is computationally expensive and presents several challenges. In particular, the inner integral of the variational form associated with the stiffness matrix is defined over the intersections of FE mesh elements with a ball of radius δ, where δ is the range of nonlocal interaction. Identifying and parameterizing these intersections is a nontrivial computational geometry problem. In this work, we propose a quadrature technique where the inner integration is performed using quadrature points distributed over the full ball, without regard for how it intersects elements, and weights are computed based on the generalized moving least squares method. Thus, as opposed to all previously employed methods, our technique does not require element-by-element integration and fully circumvents the computation of element–ball intersections. This paper considers one- and two-dimensional implementations of piecewise linear continuous FE approximations, focusing on the case where the element size h and the nonlocal radius δ are proportional, as is typical of practical computations. When boundary conditions are treated carefully and the outer integral of the variational form is computed accurately, the proposed method is asymptotically compatible in the limit of h∼δ→0, featuring at least first-order convergence in L2 for all dimensions, using both uniform and nonuniform grids. Moreover, in the case of uniform grids, the proposed method passes a patch test and, according to numerical evidence, exhibits an optimal, second-order convergence rate. Our numerical tests also indicate that, even for nonuniform grids, second-order convergence can be observed over a substantial pre-asymptotic regime. © 2022 Elsevier B.V. [less ▲] Detailed reference viewed: 27 (3 UL)A hyper-reduction method using adaptivity to cut the assembly costs of reduced order models Hale, Jack ; ; Baroli, Davide et al in Computer Methods in Applied Mechanics and Engineering (2021), 380 At every iteration or timestep of the online phase of some reduced-order modelling schemes, large linear systems must be assembled and then projected onto a reduced order basis of small dimension. The ... [more ▼] At every iteration or timestep of the online phase of some reduced-order modelling schemes, large linear systems must be assembled and then projected onto a reduced order basis of small dimension. The projected small linear systems are cheap to solve, but assembly and projection are now the dominant computational cost. In this paper we introduce a new hyper-reduction strategy called reduced assembly (RA) that drastically cuts these costs. RA consists of a triangulation adaptation algorithm that uses a local error indicator to con- struct a reduced assembly triangulation specially suited to the reduced order basis. Crucially, this reduced assembly triangulation has fewer cells than the original one, resulting in lower assembly and projection costs. We demonstrate the efficacy of RA on a Galerkin-POD type reduced order model (RAPOD). We show performance increases of up to five times over the baseline Galerkin-POD method on a non-linear reaction-diffusion problem solved with a semi-implicit time-stepping scheme and up to seven times for a 3D hyperelasticity problem solved with a continuation Newton-Raphson algorithm. The examples are implemented in the DOLFIN finite element solver using PETSc and SLEPc for linear algebra. Full code and data files to produce the results in this paper are provided as supplementary material. [less ▲] Detailed reference viewed: 423 (54 UL)Accelerating fatigue simulations of a phase-field damage model for rubber Loew, Pascal Juergen ; ; Peters, Bernhard et al in Computer Methods in Applied Mechanics and Engineering (2020), 370(113247), Phase-field damage models are able to describe crack nucleation as well as crack propagation and coalescence without additional technicalities, because cracks are treated in a continuous, spatially finite ... [more ▼] Phase-field damage models are able to describe crack nucleation as well as crack propagation and coalescence without additional technicalities, because cracks are treated in a continuous, spatially finite manner. Previously, we have developed a phase-field model to capture the rate-dependent failure of rubber, and we have further enhanced it to describe failure due to cyclic loading. Although the model accurately describes fatigue failure, the associated cyclic simulations are slow. Therefore, this contribution presents an acceleration scheme for cyclic simulations of our previously introduced phase-field damage model so that the simulation speed is increased to facilitate large-scale simulations of industrially relevant problems. We formulate an explicit and an implicit cycle jump method, which, depending on the selected jump size, reduce the calculation time up to 99.5%. To circumvent the manual tuning of the jump size, we also present an adaptive jump size selection procedure. Thanks to the implicit adaptive scheme, all material parameters are identified from experiments, which include fatigue crack nucleation and crack growth. Finally, the model and its parameters are validated with additional measurements of the fatigue crack growth rate. [less ▲] Detailed reference viewed: 165 (4 UL)Generalized quasicontinuum modeling of metallic lattices with geometrical and material nonlinearity and variability Chen, Li ; Beex, Lars ; et al in Computer Methods in Applied Mechanics and Engineering (2020), 366(112878), We propose a generalized quasicontinuum method to model the mechanical response of 3D lattice structures. The method relies on the spatial coupling of fully-resolved domains and coarse-grained domains. In ... [more ▼] We propose a generalized quasicontinuum method to model the mechanical response of 3D lattice structures. The method relies on the spatial coupling of fully-resolved domains and coarse-grained domains. In the fully-resolved domain, the full micro-structure is taken into account. In the coarse-grained domain, the kinematics of the micro-structure are individually interpolated based on their connectivity. On top of that, the contributions of the microstructure to the governing equations in the coarse-grained domain are sampled using only a few unit cells. In both domains, geometrical and material variability along the strut can be naturally taken into account using a 3D co-rotational beam finite element with embedded plastic hinges. We verify the approach for BCC lattices, demonstrating that the new method can capture both material and geometrical non-linearities of single struts at a fraction of the cost of a direct numerical simulation. [less ▲] Detailed reference viewed: 111 (16 UL)Structural shape optimization of three dimensional acoustic problems with isogeometric boundary element methods ; Lian, Haojie ; et al in Computer Methods in Applied Mechanics and Engineering (2019), 355 The boundary element method (BEM) is a powerful tool in computational acoustics, because the analysis is conducted only on structural surfaces, compared to the finite element method (FEM) which resorts to ... [more ▼] The boundary element method (BEM) is a powerful tool in computational acoustics, because the analysis is conducted only on structural surfaces, compared to the finite element method (FEM) which resorts to special techniques to truncate infinite domains. The isogeometric boundary element method (IGABEM) is a recent progress in the category of boundary element approaches, which is inspired by the concept of isogeometric analysis (IGA) and employs the spline functions of CAD as basis functions to discretize unknown physical fields. As a boundary representation approach, IGABEM is naturally compatible with CAD and thus can directly perform numerical analysis on CAD models, avoiding the cumbersome meshing procedure in conventional FEM/BEM and eliminating the difficulty of volume parameterization in isogeometric finite element methods. The advantage of tight integration of CAD and numerical analysis in IGABEM renders it particularly attractive in the application of structural shape optimization because (1) the geometry and the analysis can be interacted, (2) remeshing with shape morphing can be avoided, and (3) an optimized solution returns a CAD geometry directly without postprocessing steps. In the present paper, we apply the IGABEM to structural shape optimization of three dimensional exterior acoustic problems, fully exploiting the strength of IGABEM in addressing infinite domain problems and integrating CAD and numerical analysis. We employ the Burton–Miller formulation to overcome fictitious frequency problems, in which hyper-singular integrals are evaluated explicitly. The gradient-based optimizer is adopted and shape sensitivity analysis is conducted with implicit differentiation methods. The design variables are set to be the positions of control points which directly determine the shape of structures. Finally, numerical examples are provided to verify the algorithm. [less ▲] Detailed reference viewed: 103 (0 UL)Computational chemo-thermo-mechanical coupling phase-field model for complex fracture induced by early-age shrinkage and hydration heat in cement-based materials Nguyen, Thanh Tung ; Waldmann, Danièle ; in Computer Methods in Applied Mechanics and Engineering (2019), 348 In this paper, we present a new multi-physics computational framework that enables us to capture and investigate complex fracture behavior in cement-based materials at early-age. The present model ... [more ▼] In this paper, we present a new multi-physics computational framework that enables us to capture and investigate complex fracture behavior in cement-based materials at early-age. The present model consists of coupling the most important chemo-thermo-mechanical processes to describe temperature evolution, variation of hydration degree, and mechanical behavior. The changes of material properties are expressed as a function of the hydration degree, to capture the age effects. Fracture analysis of these processes are then accommodated by a versatile phase field model in the framework of smeared crack models, addressing the influence of cracks on hydration and thermal transfer. We additionally describe a stable and robust numerical algorithm, which aims to solve coupled problems by using a staggered scheme. The developed approach is applied to study the fracture phenomena at both macroscopic and mesoscopic scales, in which all microstructural heterogeneities of sand and cement matrix are explicitly accounted. Nucleation, initiation, and propagation of complex crack network are simulated in an efficient way demonstrating the potential of the proposed approach to assess the early-age defects in concrete structures and materials. [less ▲] Detailed reference viewed: 241 (29 UL)A unified enrichment approach addressing blending and conditioning issues in enriched finite elements ; ; Bordas, Stéphane in Computer Methods in Applied Mechanics and Engineering (2019), 349 We present a combination of techniques to improve the convergence and conditioning properties of partition of unity (PU) enriched finite element methods. By applying these techniques to different types of ... [more ▼] We present a combination of techniques to improve the convergence and conditioning properties of partition of unity (PU) enriched finite element methods. By applying these techniques to different types of enrichment functions, namely polynomial, discontinuous and singular, higher order convergence rates can be obtained while keeping condition number growth rates similar to the ones corresponding to standard finite elements. [less ▲] Detailed reference viewed: 98 (0 UL)Model order reduction accelerated Monte Carlo stochastic isogeometric method for the analysis of structures with high-dimensional and independent material uncertainties Ding, Chensen ; ; et al in Computer Methods in Applied Mechanics and Engineering (2019), 349 Structural stochastic analysis is vital to engineering. However, current material related uncertainty methods are mostly limited to low dimension, and they mostly remain unable to account for spatially ... [more ▼] Structural stochastic analysis is vital to engineering. However, current material related uncertainty methods are mostly limited to low dimension, and they mostly remain unable to account for spatially uncorrelated material uncertainties. They are not representative of realistic and practical engineering situations. In particular, it is more serious for composite structures comprised of dissimilar materials. Therefore, we propose a novel model order reduction via proper orthogonal decomposition accelerated Monte Carlo stochastic isogeometric method (IGA-POD-MCS) for stochastic analysis of exactly represented (composite) structures. This approach particularly enables high-dimensional material uncertainties wherein the characteristics of each element are independent. And the novelties include: (1) the structural geometry is exactly modeled thanks to isogeometric analysis (IGA), as well as providing more accurate deterministic and stochastic solutions, (2) we innovatively consider high-dimensional and independent material uncertainties by separating the stochastic mesh from the IGA mesh, and modeling different stochastic elements to have different (independent) uncertainty behaviors, (3) the classical Monte Carlo simulation (MCS) is employed to universally solve the high-dimensional uncertainty problem. However, to circumvent its computational expense, we employ model order reduction via proper orthogonal decomposition (POD) into the IGA coupled MCS stochastic analysis. In particular, we observe that this work decouples all IGA elements and hence permits independent uncertainty models easily, thereby the engineering problem is modeled to be more realistic and authentic. Several illustrative numerical examples verify the proposed IGA-POD-MCS approach is effective and efficient; and the larger the scale of the problem is, the more advantageous the method will become. [less ▲] Detailed reference viewed: 57 (2 UL)A simple and robust computational homogenization approach for heterogeneous particulate composites ; ; et al in Computer Methods in Applied Mechanics and Engineering (2019), 349 In this article, a computationally efficient multi-split MsXFEM is proposed to evaluate the elastic properties of heterogeneous materials. The multi-split MsXFEM is the combination of multi-split XFEM ... [more ▼] In this article, a computationally efficient multi-split MsXFEM is proposed to evaluate the elastic properties of heterogeneous materials. The multi-split MsXFEM is the combination of multi-split XFEM with multiscale finite element methods (MsFEM). The multi-split XFEM is capable to model multiple discontinuities in a single element which leads to reduction in the number of mesh elements, whereas MsFEM helps in reducing the computational time. Strain energy based homogenization has been implemented on an RVE (having volume fraction of heterogeneities up to 50%) for evaluating the elastic properties. From macro-element size analysis, we estimate that the RVE edge length must be 5 times the edge length of the macro-element. The directional analysis has been performed to verify the isotropic behavior of the material, whereas contrast analysis has been done to check the numerical accuracy of the proposed scheme. A level set correction (LSC) based on higher order shape functions has been proposed to reduce mapping errors of level set values. It is also observed that multi-split MsXFEM is about 16 times computationally more efficient than MsXFEM for 50% volume of heterogeneities. [less ▲] Detailed reference viewed: 76 (0 UL)Fracture modeling with the adaptive XIGA based on locally refined B-splines ; ; et al in Computer Methods in Applied Mechanics and Engineering (2019), 354 Detailed reference viewed: 101 (2 UL)A volume-averaged nodal projection method for the Reissner-Mindlin plate model ; ; Hale, Jack et al in Computer Methods in Applied Mechanics and Engineering (2018), 341 We introduce a novel meshfree Galerkin method for the solution of Reissner-Mindlin plate problems that is written in terms of the primitive variables only (i.e., rotations and transverse displacement) and ... [more ▼] We introduce a novel meshfree Galerkin method for the solution of Reissner-Mindlin plate problems that is written in terms of the primitive variables only (i.e., rotations and transverse displacement) and is devoid of shear-locking. The proposed approach uses linear maximum-entropy approximations and is built variationally on a two-field potential energy functional wherein the shear strain, written in terms of the primitive variables, is computed via a volume-averaged nodal projection operator that is constructed from the Kirchhoff constraint of the three-field mixed weak form. The stability of the method is rendered by adding bubble-like enrichment to the rotation degrees of freedom. Some benchmark problems are presented to demonstrate the accuracy and performance of the proposed method for a wide range of plate thicknesses. [less ▲] Detailed reference viewed: 215 (21 UL)Corotational cut finite element method for real-time surgical simulation: Application to needle insertion simulation Bui, Huu Phuoc ; Tomar, Satyendra ; Bordas, Stéphane in Computer Methods in Applied Mechanics and Engineering (2018), 345 We present the corotational cut Finite Element Method (FEM) for real-time surgical simulation. The only requirement of the proposed method is a background mesh, which is not necessarily conforming to the ... [more ▼] We present the corotational cut Finite Element Method (FEM) for real-time surgical simulation. The only requirement of the proposed method is a background mesh, which is not necessarily conforming to the boundaries/interfaces of the simulated object. The details of the surface, which can be directly obtained from binary images, are taken into account by a multilevel embedding algorithm which is applied to elements of the background mesh that are cut by the surface. Dirichlet boundary conditions can be implicitly imposed on the surface using Lagrange multipliers, whereas traction or Neumann boundary conditions, which is/are applied on parts of the surface, can be distributed to the background nodes using shape functions. The implementation is verified by convergences studies, of the geometry and of numerical solutions, which exhibit optimal rates. To verify the reliability of the method, it is applied to various needle insertion simulations (e.g. for biopsy or brachytherapy) into brain and liver models. The numerical results show that, while retaining the accuracy of the standard FEM, the proposed method can (1) make the discretisation independent from geometric description, (2) avoid the complexity of mesh generation for complex geometries, and (3) provide computational speed suitable for real-time simulations. Thereby, the proposed method is very suitable for patient-specific simulations as it improves the simulation accuracy by automatically, and properly, taking the simulated geometry into account, while keeping the low computational cost. [less ▲] Detailed reference viewed: 74 (0 UL)Improving the conditioning of XFEM/GFEM for fracture mechanics problems through enrichment quasi-orthogonalization ; Bordas, Stéphane ; in Computer Methods in Applied Mechanics and Engineering (2018) Partition of unity enrichment is known to significantly enhance the accuracy of the finite element method by allowing the incorporation of known characteristics of the solution in the approximation space ... [more ▼] Partition of unity enrichment is known to significantly enhance the accuracy of the finite element method by allowing the incorporation of known characteristics of the solution in the approximation space. However, in several cases it can further cause conditioning problems for which a number of remedies have been proposed in the framework of the extended/generalized finite element method (XFEM/GFEM). Those solutions often involve significant modifications to the initial method and result in increased implementation complexity. In the present work, a simple procedure for the local near-orthogonalization of enrichment functions is introduced, which significantly improves the conditioning of the resulting system matrices, while requiring only minor modifications to the initial method. Although application to different types of enrichment functions is possible, the resulting scheme is specialized for the singular enrichment functions used in linear elastic fracture mechanics and tested through benchmark problems. [less ▲] Detailed reference viewed: 180 (3 UL)Skew-symmetric Nitsche’s formulation in isogeometric analysis: Dirichlet and symmetry conditions, patch coupling and frictionless contact ; ; et al in Computer Methods in Applied Mechanics and Engineering (2018), 341 A simple skew-symmetric Nitsche’s formulation is introduced into the framework of isogeometric analysis (IGA) to deal with various problems in small strain elasticity: essential boundary conditions ... [more ▼] A simple skew-symmetric Nitsche’s formulation is introduced into the framework of isogeometric analysis (IGA) to deal with various problems in small strain elasticity: essential boundary conditions, symmetry conditions for Kirchhoff plates, patch coupling in statics and in modal analysis as well as Signorini contact conditions. For linear boundary or interface conditions, the skew-symmetric formulation is parameter-free. For contact conditions, it remains stable and accurate for a wide range of the stabilization parameter. Several numerical tests are performed to illustrate its accuracy, stability and convergence performance. We investigate particularly the effects introduced by Nitsche’s coupling, including the convergence performance and condition numbers in statics as well as the extra “outlier” frequencies and corresponding eigenmodes in structural dynamics. We present the Hertz test, the block test, and a 3D self-contact example showing that the skew-symmetric Nitsche’s formulation is a suitable approach to simulate contact problems in IGA. [less ▲] Detailed reference viewed: 198 (1 UL)A parallel and efficient multi-split XFEM for 3-D analysis of heterogeneous materials ; ; et al in Computer Methods in Applied Mechanics and Engineering (2018) We propose a parallel and computationally efficient multi-split XFEM approach for 3-D analysis of heterogeneous materials. In this approach, multiple discontinuities (pores and reinforcement particles ... [more ▼] We propose a parallel and computationally efficient multi-split XFEM approach for 3-D analysis of heterogeneous materials. In this approach, multiple discontinuities (pores and reinforcement particles) may intersect any given element (we call those elements multi-split elements). These discontinuities are modeled by imposing additional degrees of freedom at the nodes. The main advantage of the proposed scheme is that the mesh size remains independent of the relative distance among the heterogeneities/discontinuities. The pores and reinforcement particles are assumed to be spherical. The simulations are performed for uniform and non-uniform heterogeneity distribution. The Young’s modulus of the heterogeneous material is evaluated for different amount of pores and reinforcement particles. To demonstrate the computational efficiency of the multi-split XFEM, elastic damage analysis is performed for the unit cell with 5% pores and 5% reinforcement particles under uniaxial tensile loading. These simulations show that the Young’s modulus decreases linearly with the increase in the volume fraction of the pores and increases linearly with the increase in volume fraction of reinforcement particles. The multi-split XFEM is found to be at least 1.8 times computationally efficient than standard XFEM and at least 6.7 times computationally efficient than FEM. [less ▲] Detailed reference viewed: 147 (2 UL)Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization ; ; et al in Computer Methods in Applied Mechanics and Engineering (2018), 328 In this paper, we propose a general framework for constructing IGA-suitable planar B-spline parameterizations from given complex CAD boundaries. Instead of the computational domain bounded by four B ... [more ▼] In this paper, we propose a general framework for constructing IGA-suitable planar B-spline parameterizations from given complex CAD boundaries. Instead of the computational domain bounded by four B-spline curves, planar domains with high genus and more complex boundary curves are considered. Firstly, some pre-processing operations including B´ezier extraction and subdivision are performed on each boundary curve in order to generate a high-quality planar parameterization; then a robust planar domain partition framework is proposed to construct high-quality patch-meshing results with few singularities from the discrete boundary formed by connecting the end points of the resulting boundary segments. After the topology information generation of quadrilateral decomposition, the optimal placement of interior B´ezier curves corresponding to the interior edges of the quadrangulation is constructed by a global optimization method to achieve a patch-partition with high quality. Finally, after the imposition of C1/G1-continuity constraints on the interface of neighboring Bezier patches with respect to each quad in the quadrangulation, the high-quality Bezier patch parameterization is obtained by a local optimization method to achieve uniform and orthogonal iso-parametric structures while keeping the continuity conditions between patches. The efficiency and robustness of the proposed method are demonstrated by several examples which are compared to results obtained by the skeleton-based parameterization approach. [less ▲] Detailed reference viewed: 153 (7 UL)Adaptive Isogeometric analysis for plate vibrations: An efficient approach of local refinement based on hierarchical a posteriori error estimation ; ; Tomar, Satyendra et al in Computer Methods in Applied Mechanics and Engineering (2018), 342 This paper presents a novel methodology of local adaptivity for the frequency-domain analysis of the vibrations of Reissner–Mindlin plates. The adaptive discretization is based on the recently developed ... [more ▼] This paper presents a novel methodology of local adaptivity for the frequency-domain analysis of the vibrations of Reissner–Mindlin plates. The adaptive discretization is based on the recently developed Geometry Independent Field approximaTion (GIFT) framework, which may be seen as a generalization of the Iso-Geometric Analysis (IGA).Within the GIFT framework, we describe the geometry of the structure exactly with NURBS (Non-Uniform Rational B-Splines), whilst independently employing Polynomial splines over Hierarchical T-meshes (PHT)-splines to represent the solution field. The proposed strategy of local adaptivity, wherein a posteriori error estimators are computed based on inexpensive hierarchical h-refinement, aims to control the discretization error within a frequency band. The approach sweeps from lower to higher frequencies, refining the mesh appropriately so that each of the free vibration mode within the targeted frequency band is sufficiently resolved. Through several numerical examples, we show that the GIFT framework is a powerful and versatile tool to perform local adaptivity in structural dynamics. We also show that the proposed adaptive local h-refinement scheme allows us to achieve significantly faster convergence rates than a uniform h-refinement. [less ▲] Detailed reference viewed: 171 (5 UL)Adaptive multi-patch isogeometric analysis based on locally refined B-splines ; ; et al in Computer Methods in Applied Mechanics and Engineering (2018), 339 Detailed reference viewed: 85 (1 UL)Accelerating Monte Carlo estimation with derivatives of high-level finite element models Hauseux, Paul ; Hale, Jack ; Bordas, Stéphane in Computer Methods in Applied Mechanics and Engineering (2017), 318 In this paper we demonstrate the ability of a derivative-driven Monte Carlo estimator to accelerate the propagation of uncertainty through two high-level non-linear finite element models. The use of ... [more ▼] In this paper we demonstrate the ability of a derivative-driven Monte Carlo estimator to accelerate the propagation of uncertainty through two high-level non-linear finite element models. The use of derivative information amounts to a correction to the standard Monte Carlo estimation procedure that reduces the variance under certain conditions. We express the finite element models in variational form using the high-level Unified Form Language (UFL). We derive the tangent linear model automatically from this high-level description and use it to efficiently calculate the required derivative information. To study the effectiveness of the derivative-driven method we consider two stochastic PDEs; a one- dimensional Burgers equation with stochastic viscosity and a three-dimensional geometrically non-linear Mooney-Rivlin hyperelastic equation with stochastic density and volumetric material parameter. Our results show that for these problems the first-order derivative-driven Monte Carlo method is around one order of magnitude faster than the standard Monte Carlo method and at the cost of only one extra tangent linear solution per estimation problem. We find similar trends when comparing with a modern non-intrusive multi-level polynomial chaos expansion method. We parallelise the task of the repeated forward model evaluations across a cluster using the ipyparallel and mpi4py software tools. A complete working example showing the solution of the stochastic viscous Burgers equation is included as supplementary material. [less ▲] Detailed reference viewed: 1929 (239 UL) |
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