A direct simulation method and lower-bound estimation for a class of gamma random fields with applications in modelling material properties

Liu, Yong; Shields, Michael D.

doi:10.1016/j.probengmech.2017.01.001

Cited by 25 publications

(7 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Afterwards, the consistent tangent stiffness matrix is computed [24]. This matrix is then used to compute the updated element stiffness matrix (15), resulting in an updated system stiffness matrix K.…”

Section: The Finite Element Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Multilevel Monte Carlo for uncertainty quantification in structural engineering

Blondeel,

Robbe,

van hoorickx

et al. 2018

Preprint

View full text Add to dashboard Cite

Practical structural engineering problems often exhibit a significant degree of uncertainty in the material properties being used, the dimensions of the modeled structures, the magnitude of loading forces, etc. In this paper, we consider two beam models: a cantilever beam clamped at one end and a beam clamped at both ends. We consider a static and a dynamic load. The material uncertainty resides in the Young's modulus, which is either modeled by means of one random variable sampled from a univariate Gamma distribution or with multiple random variables sampled from a Gamma random field. The Gamma random field is generated starting from a truncated Karhunen-Loève expansion of a Gaussian random field, followed by a transformation. Three different responses are considered: the static elastic response, the dynamic elastic response and the static elastoplastic response. The first two respectively simulate the spatial displacement of a concrete beam and its frequency response function in the elastic domain. The third one simulates the spatial displacement of a steel beam in the elastic as well as in the plastic domain. The plastic region is governed by the von Mises yield criterion with linear isotropic hardening. In order to compute the statistical quantities of the static deflection and frequency response function, Multilevel Monte Carlo (MLMC) is combined with a Finite Element solver. This recent sampling method is based on the idea of variance reduction, and employs a hierarchy of finite element discretizations of the structural engineering model. The good performance of MLMC arises from its ability to take many computationally cheap samples on the coarser meshes of the hierarchy, and only few computationally expensive samples on the finer meshes. In this paper, the computational costs and run times of the MLMC method are compared with those of the classical Monte Carlo method, demonstrating a significant speedup of up to several orders of magnitude for the studied cases. For the static elastic response, the deflection of the beam including uncertainty bounds in the spatial domain is visualized. For the static elastoplastic response, the focus lies on the visualization of a force deflection curve including uncertainty bounds. For the dynamic elastic response, the visualizations consist of a frequency response function, including uncertainty bounds.

show abstract

Section: The Finite Element Methodsmentioning

confidence: 99%

“…Following [15], we opt to describe the Young's modulus in the homogeneous model by means of a univariate Gamma distribution. This distribution is characterized by a shape parameter α and a scale parameter β:…”

Section: The Homogeneous Modelmentioning

confidence: 99%

Multilevel Monte Carlo for uncertainty quantification in structural engineering

Blondeel,

Robbe,

van hoorickx

et al. 2018

Preprint

View full text Add to dashboard Cite

show abstract

“…Following [16], we opt to describe the Young's modulus in the homogeneous model by means of a univariate gamma distribution. This distribution is characterized by a shape parameter α and a scale parameter β, and its probability density function given by…”

Section: The Homogeneous Modelmentioning

confidence: 99%

h- and p-refined Multilevel Monte Carlo Methods for Uncertainty Quantification in Structural Engineering

Blondeel,

Robbe,

van hoorickx

et al. 2019

Preprint

View full text Add to dashboard Cite

Practical structural engineering problems are often characterized by significant uncertainties. Historically, one of the prevalent methods to account for this uncertainty has been the standard Monte Carlo (MC) method. Recently, improved sampling methods have been proposed, based on the idea of variance reduction by employing a hierarchy of mesh refinements. We combine an h-and p-refinement hierarchy with the Multilevel Monte Carlo (MLMC) and Multilevel Quasi-Monte Carlo (MLQMC) method. We investigate the applicability of these novel combination methods on three structural engineering problems, for which the uncertainty resides in the Young's modulus: the static response of a cantilever beam with elastic material behavior, its static response with elastoplastic behavior, and its dynamic response with elastic behavior.The uncertainty is either modeled by means of one random variable sampled from a univariate Gamma distribution or with multiple random variables sampled from a gamma random field. This random field results from a truncated Karhunen-Loève (KL) expansion. In this paper, we compare the computational costs of these Monte Carlo methods. We demonstrate that MQLMC and MLMC have a significant speedup with respect to MC, regardless of the mesh refinement hierarchy used. We empirically demonstrate that the MLQMC cost is optimally proportional to −1 under certain conditions, where is the tolerance on the root-mean-square error (RMSE). In addition, we show that, when the uncertainty is modeled as a random field, the multilevel methods combined with p-refinement have a significant lower computation cost than their counterparts based on h-refinement. We also illustrate the effect the uncertainty models have on the uncertainty bounds in the solutions. An uncertain Young's modulus modeled as a single random variable has much larger uncertainty bounds on its solution than an uncertain Young's modulus modeled as a random field.

show abstract

“…There is a wide range of methods for simulation of Gaussian or non-Gaussian random processes and fields [18][19][20][21][22][23][24][25][26]. The problem is that most of the methods could be used only for simulation of some…”

Section: Introductionmentioning

confidence: 99%

Stochastic Model of Spatial Fields of the Average Daily Wind Chill Index

Kargapolova

2020

Information

View full text Add to dashboard Cite

The objective of this paper was to construct a numerical stochastic model of the spatial field of the average daily wind chill index on an irregular grid defined by the location of the weather stations. It is shown in the paper that the field in question was heterogeneous and non-Gaussian. A stochastic model based on the real data collected at the weather stations located in West Siberia and on the method of the inverse distribution function that sufficiently well reproduce different characteristics of the real field of the average daily wind chill index is proposed in this paper. I also discussed several questions related to the simulation of the field on a regular grid. In the future, my intention is to transform the model proposed to a model of the conditional spatio-temporal field defined on a regular grid that allows one to forecast the wind chill index.

show abstract

A direct simulation method and lower-bound estimation for a class of gamma random fields with applications in modelling material properties

Cited by 25 publications

References 26 publications

Multilevel Monte Carlo for uncertainty quantification in structural engineering

Multilevel Monte Carlo for uncertainty quantification in structural engineering

h- and p-refined Multilevel Monte Carlo Methods for Uncertainty Quantification in Structural Engineering

Stochastic Model of Spatial Fields of the Average Daily Wind Chill Index

Contact Info

Product

Resources

About