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In the past two decades, Europe has witnessed a significant transition in the design codes used for assessing foundation structures, with the widespread adoption of the Eurocodes (EC). This shift remains a pertinent topic within the engineering community, particularly concerning the transition from traditional design methodologies to those prescribed by the Eurocodes, as well as the potential for fully probabilistic design. While the Eurocodes’ methodology is described as probabilistic, it is crucial to recognize that the achievement of the target reliability level is predominantly facilitated through a system of partial safety factors. These factors are integrated into the calculation algorithm as fixed values, rendering the process essentially deterministic. To refine these calculations for more accurate reliability estimates—expressed in terms of failure probability—a genuinely probabilistic framework is required, termed as fully probabilistic computation. This paper aims to elucidate the fully probabilistic calculation approach for the broader professional community, using the geotechnical application of shallow foundations as an illustrative example. We present a comparative analysis of this advanced approach with the standard foundation design according to EC7 and ČSN 731001, the latter being a precursor in Europe for implementing the partial safety factor method. The discussion extends to a practical demonstration of full probabilistic design juxtaposed against the conventional partial safety factor method, using a shallow foundation case study. Furthermore, the paper delves into the impact of the tail behavior of uncertain or spatially varying soil parameters on the theoretical probability of failure, underscoring its significance in foundation design. [ABSTRACT FROM AUTHOR]

The paper introduces a practical method for the construction of large-scale point sets for analysis of computer models. The constructed experimental design is useful for (probabilistic) integration, construction of approximation or a screening. The essence of the presented approach is the stratification of the design domain into an orthogonal grid of substrata and a subsequent tiling with tiles of points. If optimized, such tiles experience a major reduction of the number of degrees of freedom in the optimization process. That way, optimal or near-optimal point patterns can be feasibly identified and are further utilized for construction of larger point sets, thanks to the idea of self-similarity and structured space stratification The space-filling properties of the resulting point sets may be further enhanced by various "scrambling" strategies, which may remove the undesired sample collapsibility achieved via regular tiling. The performance of the constructed point sets is compared to Quasi Monte Carlo (QMC), Randomized Quasi Monte Carlo (RQMC) sequences, which are still today considered by engineers and even scientists as choices for variance reduction of numerical integration Further, the mentioned sampling strategies are compared in the terms of robustness when integrating a multivariate function with a localized feature. It is concluded that the proposed sampling approach reaches a superior performance in numerical integration and identification of function extremes as compared to sampling methods used by practicing researchers and engineers. Additionally, the reader is supplied with the open-access, ready-to-use implementation of the presented algorithm named SampleTiler. • Construction of large-scale point samples by balancing the bias–variance tradeoff. • Stratification of the design domain and subsequent tiling with tiles of points. • Construction of large optimized experimental designs due to self-similarity. • Superior performance of samples as compared to commonly used methods. • Open-access Python code provided on a GitHub repository. [ABSTRACT FROM AUTHOR]

This article proposes a sampling technique that delivers robust designs, that is, point sets selected from a design domain in the shape of a unit hypercube. The designs are guaranteed to provide a statistically uniform point distribution, meaning that every location has the same probability of being selected. Moreover, the designs are sample uniform, meaning that each individual design has its points spread evenly throughout the domain. The sample uniformity (often measured via a discrepancy criterion) is achieved using distance-based criteria ( ϕ p or Maximin), that is, criteria normally used in space-filling designs. We show that the standard intersite metrics employed in distance-based criteria (Maximin and ϕ p (phi)) do not deliver statistically uniform designs. Similarly, designs optimized via centered L2 discrepancy or support points are also not statistically uniform. When these designs (after optimization based on intersite distances) are used for Monte Carlo type of integration, their statistical nonuniformity is a serious problem as it may lead to a systematic bias. This article proposes using a periodic metric to guarantee the statistical uniformity of the family of distance-based designs. The presented designs used as benchmarks in the article are only taken from the class of Latin hypercube designs, which forces univariate projections to be uniform and improves accuracy in Monte Carlo integration of some functions. for this article are available online. [ABSTRACT FROM AUTHOR]

We propose an algorithm for selection of points from the design domain of small to moderate dimension and for failure probability estimation. The proposed active learning detects failure events and progressively refines the boundary between safe and failure domains thereby improving the failure probability estimation. The method is particularly useful when each evaluation of the performance function g (x) is very expensive and the function can be characterized as either highly nonlinear, noisy, or even discrete-state (e.g., binary). In such cases, only a limited number of calls is feasible, and gradients of g (x) cannot be used. The input design domain is progressively segmented by expanding and adaptively refining a mesh-like lock-free geometrical structure. The proposed triangulation-based approach effectively combines the features of simulation and approximation methods. The algorithm performs two independent tasks: (i) the estimation of probabilities through an ingenious combination of deterministic cubature rules and the application of the divergence theorem and (ii) the sequential extension of the experimental design with new points. The sequential selection of points from the design domain for future evaluation of g (x) is carried out through a new decision approach, which maximizes instantaneous information gain in terms of the probability classification that corresponds to the local region. The extension may be halted at any time, e.g., when sufficiently accurate estimations are obtained. Due to the use of the exact geometric representation in the input domain, the algorithm is most effective for problems of a low dimension, not exceeding eight. The method can handle random vectors with correlated non-Gaussian marginals. When the values of the performance function are valid and credible, the estimation accuracy can be improved by employing a smooth surrogate model based on the evaluated set of points. Finally, we define new factors of global sensitivity to failure based on the entire failure surface weighted by the density of the input random vector. [Display omitted] • A novel adaptive sequential extension of the sample by building a mesh-like geometrical structure. • A very small number of evaluations of the expensive model, which may return a categorical output only. • Robust sequential design extension optimally balances between exploitation and exploration. • An accurate estimation of probabilities via the divergence theorem and cubature rules. • Novel global sensitivities to failure based on the entire failure surface, weighted by probability density. [ABSTRACT FROM AUTHOR]

Highlights • Rasterized approximation of hyper-dimensional Voronoi diagram. • Exctraction of selected scalar topological attributes of cells. • Efficient massively parallel solution providing significant speedup in comparison with full analytic solution. • Explicit estimation of rasterization error in general dimension. Abstract This paper concerns an algorithm for fast parallel approximation of selected attributes of hyper-dimensional Voronoï cells in a unit hypercube. The presented algorithm does not require the construction of a corresponding Voronoï diagram (usually by employing the Quick Hull algorithm) which typically is a highly computationally demanding task, especially when performed in higher dimensions. The algorithm is suitable for both the clipped and periodic variants of Voronoï tessellation and provides a significant speedup in a convenient range of practical usage. For the purposes of approximation of selected scalar properties of Voronoï cells, only the distances of points in sample are evaluated using an adequately fine underlying orthogonal mesh. The algorithm estimates e.g. the volumes and centroids of Voronoï cells, radii and coordinates of centers of the corresponding Delaunay hyper-circles. The paper also provides quite accurate explicit error estimators for the extracted attributes due to rasterization and thus the user is given a control over the accuracy by selecting an appropriate discretization density. Among numerous fields in which Voronoï diagrams are being utilized, the authors are concerned with optimization of point samples for the Design of Experiments and also with weighting of integration points in Monte Carlo type integration. In these applications, selected scalar topological descriptors of Voronoï diagrams are being repeatedly computed. As the full Voronoï diagram is not of interest but the resulting cell volumes or shape descriptions have to be repeatedly computed, the presented parallel solution seems highly suitable for these applications. [ABSTRACT FROM AUTHOR]

Highlights • Efficient parallel implementation of multi-dimensional particle system using GPGPU. • Analogy between potential energy of particle system and point distribution evaluation in Design of Experiments. • Periodically repeated design domain. • Generalized utilization of on-chip memory. • Efficient serialization of thread writing requests. Abstract The presented paper deals with possible approaches to parallel implementation of solution of a hyper-dimensional dynamical particle system. The proposed implementation approaches are generally applicable for similar particle systems of interest in various research and engineering fields. The original motivation for the present work was a simulation of particles that represent a space-filling design to be optimized for further use in design of experiments. Due to the underlying purpose of this particle system, the dimension of the particle system of interest is considered to be entirely arbitrary. Such a hyper-dimensional space is further folded into a periodically repeated domain. The theoretical background of the proposed particle system is provided along with the derivation of equations of motion of the dynamical system. As the complexity of the system is not limited by the number of particles nor the number of dimensions, the possibilities of utilizing the GPGPU platform are more restricted in comparison with today's fast parallel implementations of common particle systems. Two distinct approaches to parallel implementation are presented, one aiming at a generalized usage of the fast on-chip resources, the other entirely relying on the GPU's on-board global memory. Despite unambiguous mutual differences in their performance, both parallel implementations deliver major speedup compared to the single-thread CPU solution as well as a better scaling of execution time when increasing both the number of particles and dimensions. [ABSTRACT FROM AUTHOR]

The Audze–Eglājs (AE) criterion was developed to achieve a uniform distribution of experimental points in a hypercube. However, the paper shows that the AE criterion provides strongly nonuniform designs due to the effect of the boundaries of the hypercube. We propose a simple remedy that lies in the assumption of periodic boundary conditions. The biased behavior of the original AE criterion and excellent performance of the modified criterion are demonstrated using simple numerical examples focused on (i) the uniformity of sampling density over the design space and, (ii) statistical sampling efficiency measured through the ability to correctly estimate the statistical parameters of functions of random variables. An engineering example of reliability calculation is presented, too. [ABSTRACT FROM AUTHOR]

The paper presents a new efficient and robust method for rare event probability estimation for computational models of an engineering product or a process returning categorical information only, for example, either success or failure. For such models, most of the methods designed for the estimation of failure probability, which use the numerical value of the outcome to compute gradients or to estimate the proximity to the failure surface, cannot be applied. Even if the performance function provides more than just binary output, the state of the system may be a non-smooth or even a discontinuous function defined in the domain of continuous input variables. This often happens because the mathematical model features non-smooth components or discontinuities (e.g., in the constitutive laws), bifurcations, or even domains in which no reasonable model response is obtained. In these cases, the classical gradient-based methods usually fail. We propose a simple yet efficient algorithm, which performs a sequential adaptive selection of points from the input domain of random variables to extend and refine a simple distance-based surrogate model. Two different tasks can be accomplished at any stage of sequential sampling: (i) estimation of the failure probability, and (ii) selection of the best possible candidate for the subsequent model evaluation if further improvement is necessary. The proposed criterion for selecting the next point for model evaluation maximizes the expected probability classified by using the candidate. Therefore, the perfect balance between global exploration and local exploitation is maintained automatically. If there are more rare events such as failure modes, the method can be generalized to estimate the probabilities of all these event types. Moreover, when the numerical value of model evaluation can be used to build a smooth surrogate, the algorithm can accommodate this information to increase the accuracy of the estimated probabilities. Lastly, we define a new simple yet general geometrical measure of the global sensitivity of the rare-event probability to individual variables, which is obtained as a by-product of the proposed refinement algorithm. [Display omitted] • Accurate failure probability estimation for strongly nonlinear failure surfaces. • The method can work with just categorical limit state function for continuous inputs. • Adaptive sequential extension of small sample via the proposed ψ learning function. • Automatic balancing between global exploration and local exploitation in input space. • Efficient importance sampling in a ring excluding the safe ball in Gaussian space. [ABSTRACT FROM AUTHOR]

A thermodynamically consistent model proposed in this work introduces a generative symbolic framework for computational components, which can capture the inelastic behavior of interfaces with 3D kinematics in response to monotonic, cyclic, and fatigue loading histories. It contributes to a long list of existing cohesive zone models by introducing novel forms of potentials for free energy, threshold function, and dissipation to deliver the following model properties: (i) the evolution equations derived from the potentials are linked in such a way that damage develops along with the cumulative measure of plastic sliding, (ii) the convex, continuous, and smooth shape of the threshold function introduced in the effective stress domain enables a general and efficient time integration that automatically handles unloading and reloading, and (iii) the non-associative flow potential includes a scalable degree of coupling between normal and tangential damage evolution. The studies performed in the paper demonstrate that these model ingredients lead to a consistent coverage of arbitrary non-proportional loading scenarios in tangential and normal directions. For monotonic loading, the presented examples illustrate the ability of the model to reproduce the shear dilatancy, pressure sensitivity including the vertex effect, stiffness recovery in compression, and abrasion due to sliding. For cyclic fatigue loading, the studies show that the model reflects the link between the hysteretic behavior and the fatigue damage evolution. This link constitutes the basis for a physically sound description of fatigue degradation along material interfaces. It manifests itself in the presented energy breakdown with distinguished fractions of energy release due to damage and plasticity ascribed to normal and tangential directions. Two calibration and validation examples using pull-out tests exposed to monotonic and cyclic loading are provided to show the applicability of the model for the characterization of the bond between steel bars and concrete. A publicly available implementation using a computer algebra system (CAS) is provided in the form of an interactive Jupyter notebook to document the reproducibility of the results. [ABSTRACT FROM AUTHOR]

• Unified least-squares-based framework for fast "nodal" and "element" strategies of mapping scalar CT voxel densities onto finite element mesh. • Approximation of density by finite element method using least squares projection. • The discontinuous variant of the finite element method is more suitable for density approximation than the continuous variant in terms of accuracy and efficiency. • The discontinuous zero-order finite element method preserves the density spectrum better than the continuous one. The spatially varying mechanical properties in finite element models of bone are most often derived from bone density data obtained via quantitative computed tomography. The key step is to accurately and efficiently map the density given in voxels to the finite element mesh. The density projection is first formulated in least-squares terms and then discretized using a continuous and discontinuous variant of the finite element method. Both discretization variants are compared with the nodal and element approaches known from the literature. In terms of accuracy in the L2 norm, energy distance and efficiency, the discontinuous zero-order variant appears to be the most advantageous. The proposed variant sufficiently preserves the spectrum of density at the edges, while keeping computational cost low. The continuous finite element method is analogous to the nodal formulation in the literature, while the discontinuous finite element method is analogous to the element formulation. The two variants differ in terms of implementation, computational cost and ability to preserve the density spectrum. These differences cannot be described and measured by known indirect methods from the literature. [ABSTRACT FROM AUTHOR]

In this article, a novel method for the extension of sample size in Latin Hypercube Sampling (LHS) is suggested. The method can be applied when an initial LH design is employed for the analysis of functions g of a random vector. The article explains how the statistical, sensitivity and reliability analyses of g can be divided into a hierarchical sequence of simulations with subsets of samples of a random vector in such a way that (i) the favorable properties of LHS are retained (the low number of simulations needed for statistically significant estimations of statistical parameters of function g with low estimation variability); (ii) the simulation process can be halted, for example, when the estimations reach a certain prescribed statistical significance. An important aspect of the method is that it efficiently simulates subsets of samples of random vectors while focusing on their correlation structure or any other objective function such as some measure of dependence, spatial distribution uniformity, discrepancy, etc. This is achieved by employing a robust algorithm based on combinatorial optimization of the mutual ordering of samples. The method is primarily intended to serve as a tool for computationally intensive evaluations of g where there is a need for pilot numerical studies, preliminary and subsequently refined estimations of statistical parameters, optimization of the progressive learning of neural networks, or during experimental design. [ABSTRACT FROM AUTHOR]

In this article, a novel method for the extension of sample size in Latin Hypercube Sampling (LHS) is suggested. The method can be applied when an initial LH design is employed for the analysis of functions g of a random vector. The article explains how the statistical, sensitivity and reliability analyses of g can be divided into a hierarchical sequence of simulations with subsets of samples of a random vector in such a way that (i) the favorable properties of LHS are retained (the low number of simulations needed for statistically significant estimations of statistical parameters of function g with low estimation variability); (ii) the simulation process can be halted, for example, when the estimations reach a certain prescribed statistical significance. An important aspect of the method is that it efficiently simulates subsets of samples of random vectors while focusing on their correlation structure or any other objective function such as some measure of dependence, spatial distribution uniformity, discrepancy, etc. This is achieved by employing a robust algorithm based on combinatorial optimization of the mutual ordering of samples. The method is primarily intended to serve as a tool for computationally intensive evaluations of g where there is a need for pilot numerical studies, preliminary and subsequently refined estimations of statistical parameters, optimization of the progressive learning of neural networks, or during experimental design. [ABSTRACT FROM AUTHOR]

The paper presents a discrete meso-scale model for fracture of concrete taking into account random spatial variability of material parameters. Beams of various sizes, with notches of various depths, are simulated numerically to study the combination of energetic and statistical size effects. A substantial part of material randomness is shown to be caused by random locations of the largest aggregates. Further randomness, due to random fluctuations of material parameters, is considered and an effect of introducing a spatially auto-correlated random field is analyzed. The results of the simulations are compared with recently published experimental data on concrete beams in three-point bending. The differences in the role of randomness in beams of various sizes, with different notch depths, are demonstrated, and differences in energy dissipation are discussed. [ABSTRACT FROM AUTHOR]

Highlights: [•] FReET software for statistical and reliability analyses has been developed. [•] Small-sample simulation techniques have been implemented. [•] Degradation modeling enables degradation module FReET-D. [•] Applications of software for reliability analysis of concrete structures are listed. [ABSTRACT FROM AUTHOR]

This paper presents a novel adaptive sequential sampling method for building Polynomial Chaos Expansion surrogate models. The technique enables one-by-one extension of an experimental design while trying to obtain an optimal sample at each stage of the adaptive sequential surrogate model construction process. The proposed sequential sampling strategy selects from a pool of candidate points by trying to cover the design domain proportionally to their local variance contribution. The proposed criterion for the sample selection balances both exploitation of the surrogate model and exploration of the design domain. The adaptive sequential sampling technique can be used in tandem with any user-defined sampling method, and here was coupled with commonly used Latin Hypercube Sampling and advanced Coherence D-optimal sampling in order to present its general performance. The obtained numerical results confirm its superiority over standard non-sequential approaches in terms of surrogate model accuracy and estimation of the output variance. • The sequential sampling reflecting the exploration and exploitation is proposed. • The accuracy of results is higher in comparison to non-sequential approach. • The proposed criterion for sequential sampling can be used with any sampling scheme. [ABSTRACT FROM AUTHOR]

The fast growing amount of high-performance textile fabrics made of carbon, glass and basalt yarns open up enormous design scope for cementitious composites, that needs to be sorted and graded using novel characterization procedures. A combined experimental and numerical characterization method of the composite tensile behavior is presented here for concrete reinforced with non-impregnated carbon textile fabrics. Because of the heterogeneous structure of the bond layer, which consists of a non-impregnated multifilament yarn partially penetrated by the cement paste, this material combination poses particularly high demands on a realistic modeling approach. It is well known, that the tensile response of strain-hardening brittle-matrix composites is primarily shaped by the multiple cracking process of the matrix. To account also for the local damage developing in the heterogeneous bond layer ahead of a crack bridge during the multiple cracking process, we combine a probabilistic crack bridge model with a generalized version of a crack tracing algorithm reflecting the random matrix strength. The resulting probabilistic multiscale multiple cracking model has been calibrated and validated using a consistent series of crack bridge and composite tensile tests by comparing the predicted and measured stress–strain curves and crack spacing histories. The efficient, flexible and realistic modeling concept can serve as a basis for the development of rapid characterization methods and more economic and reliable design rules. [ABSTRACT FROM AUTHOR]

The article proposes a method for characterizing the in situ interaction between filaments in a multifilament yarn. The stress transfer between neighboring filaments causes the reactivation of a broken filament at some distance from the break. The utilized statistical bundle models predict a change in the slope of the mean size effect curve once the specimen length becomes longer than the stress transfer length. This fact can be exploited in order to determine the stress transfer length indirectly using the yarn tensile test with appropriately chosen test lengths. The identification procedure is demonstrated using two test series of tensile tests with AR-glass and carbon yarns. [ABSTRACT FROM AUTHOR]

Abstract: In this paper an extension of Weibull theory by the introduction of a statistical length scale is presented. The classical Weibull strength theory is self-similar; a feature that can be illustrated by the fact that the strength dependence on structural size is a power law (a straight line on a double logarithmic graph). Therefore, the theory predicts unlimited strength for extremely small structures. In the paper, it is shown that such a behavior is a direct implication of the assumption that structural elements have independent random strengths. By the introduction of statistical dependence in the form of spatial autocorrelation, the size dependent strength becomes bounded at the small size extreme. The local random strength is phenomenologically modeled as a random field with a certain autocorrelation function. In such a model, the autocorrelation length plays the role of a statistical length scale. The focus is on small failure probabilities and the related probabilistic distributions of the strength of composites. The theoretical part is followed by applications in fiber bundle models, chains of fiber bundle models and the stochastic finite element method in the context of quasibrittle failure. [Copyright &y& Elsevier]

• Shape registration was used to map bone shapes onto template bone, allowing the spatial correlation structure of bone mineral density to be studied. • Non-stationary random field of bone mineral density with deterministic age dependence and sex sensitive was explored. • Energy distance between Gaussian and non-Gaussian random fields is convergent, but to non-zero value. • Average bone mineral density can be simulated with only one coefficient in Karhunen-Loève expansion with acceptable accuracy. • Interactive web application BoneGen allows to generate pelvic bone mineral density realisations. Capturing the population variability of bone properties is of paramount importance to biomedical engineering. The aim of the present paper is to describe variability and correlations in bone mineral density with a spatial random field inferred from routine computed tomography data. Random fields were simulated by transforming pairwise uncorrelated Gaussian random variables into correlated variables through the spectral decomposition of an age-detrended correlation matrix. The validity of the random field model was demonstrated in the spatiotemporal analysis of bone mineral density. The similarity between the computed tomography samples and those generated via random fields was analyzed with the energy distance metric. The random field of bone mineral density was found to be approximately Gaussian/slightly left-skewed/strongly right-skewed at various locations. However, average bone density could be simulated well with the proposed Gaussian random field for which the energy distance, i.e., a measure that quantifies discrepancies between two distribution functions, is convergent with respect to the number of correlation eigenpairs. The proposed random field model allows the enhancement of computational biomechanical models with variability in bone mineral density, which could increase the usability of the model and provides a step forward in in-silico medicine. [Display omitted] [ABSTRACT FROM AUTHOR]

The paper presents a follow-up study of numerical modeling of complicated interplay of size effects in concrete structures. The major motivation is to identify and study interplay of several scaling lengths stemming from the material, boundary conditions and geometry. Methods of stochastic nonlinear fracture mechanics are used to model the well published results of direct tensile tests of dog-bone specimens with rotating boundary conditions. Firstly, the specimens are modeled using microplane material and also fracture-plastic material laws to show that a portion of the dependence of nominal strength on structural size can be explained deterministically. However, it is clear that more sources of size effect play a part, and we consider two of them. Namely, we model local material strength using an autocorrelated random field attempting to capture a statistical part of the combined size effect, scatter inclusive. In addition, the strength drop noticeable with small specimens which was obtained in the experiments could be explained either by the presence of a weak surface layer of constant thickness (caused e.g. by drying, surface damage, aggregate size limitation at the boundary, or other irregularities) or three dimensional effects incorporated by out-of-plane flexure of specimens. The latter effect is examined by comparison of 2D and 3D models with the same material laws. All three named sources (deterministic-energetic, statistical size effects and the weak layer effect) are believed to be the sources most contributing to the observed strength size effect; the model combining all of them is capable of reproducing the measured data. The computational approach represents a marriage of advanced computational nonlinear fracture mechanics with simulation techniques for random fields representing spatially varying material properties. Using a numerical example, we document how different sources of size effects detrimental to strength can interact and result in relatively complicated quasibrittle failure processes. The presented study documents the well known fact that the experimental determination of material parameters (needed for the rational and safe design of structures) is very complicated for quasibrittle materials such as concrete. [ABSTRACT FROM AUTHOR]

Abstract: A practical framework for generating cross correlated random fields with a specified marginal distribution function, an autocorrelation function and cross correlation coefficients is presented in the paper. The approach relies on well-known series expansion methods for simulation of a Gaussian random field. The proposed method requires all cross correlated fields over the domain to share an identical autocorrelation function and the cross correlation structure between each pair of simulated fields to be simply defined by a cross correlation coefficient. Such relations result in specific properties of eigenvectors of covariance matrices of discretized field over the domain. These properties are used to decompose the eigenproblem, which must normally be solved in computing the series expansion, into two smaller eigenproblems. Such a decomposition represents a significant reduction of computational effort. Non-Gaussian components of a multivariate random field are proposed to be simulated via memoryless transformation of underlying Gaussian random fields for which the Nataf model is employed to modify the correlation structure. In this method, the autocorrelation structure of each field is fulfilled exactly while the cross correlation is only approximated. The associated errors can be computed before performing simulations and it is shown that the errors happen only in the cross correlation between distant points and that they are negligibly small in practical situations. Some comments on available techniques for simulation of underlying random variables in connection with the accuracy of basic fields’ statistics at a given sample size are made. For this purpose a simple error assessment procedure is presented. Simulated random fields can be used both for representation of spatially correlated properties of structure or random load in the stochastic finite element method (SFEM). An example of this application is related to size effect studies in the nonlinear fracture mechanics of concrete, and is used to illustrate the method. [Copyright &y& Elsevier]

• numerical integration with designs optimized via miniMax (mM) criterion is developed. • mM designs with standard metric are nonuniform which may cause biased integration. • the proposed periodic metric ensures point uniformity and unbiased robust integration. • efficient construction algorithm for mM designs (standard or periodic) is developed. The selection of points for numerical integration of the Monte Carlo type, largely used in analysis of engineering problems, is developed. It is achieved by modification of the metric in the minimax optimality criterion. The standard minimax criterion ensures the design exhibits good space-filling property and therefore reduces the variance of the estimator of the integral. We, however, show that the points are not selected with the same probability over the space of sampling probabilities: some regions are over- or under-sampled when designs are generated repetitively. This violation of statistical uniformity may lead to systematically biased integral estimators. We propose that periodic metric be considered for calculation of the minimax distance. Such periodic minimax criterion provides statistically uniform designs leading to unbiased integration results and also low estimator variance due to retained space-filling property. These conclusions are demonstrated by examples integrating analytical functions. The designs are constructed by two different algorithms: (i) a new time-stepping algorithm resembling a damped system of attracted particles developed here, and (ii) the heuristic swapping of coordinates. The designs constructed by the time-stepping algorithm are attached to the paper as a supplementary material. The computer code for construction of the designs is attached too. [ABSTRACT FROM AUTHOR]

• Major refinements of the distance-based criterion for uniform space-filling designs. • The lower bound on the exponent in ϕ p criterion for space-filling design is derived. • The improved criterion becomes useful for small-sample designs and high dimensions. • The correct shape of the interaction domain for a pair of points is proposed. • Energy and force contrast in high dimensions is derived for the Euclidean metric. In this paper, the family of ϕ p optimization criteria for space-filling designs is critically reviewed, with a focus on its behavior in moderate to large dimensions, especially for small sample sizes (low saturations of the design domain). Problems that arise during the standard use of the ϕ p criteria for the optimization of point sets in standard hypercubic design domains are identified and adequate remedies are proposed. It is shown how the distance exponent in the distance-based criteria should be dependent on the domain dimension. In cases of small sample sizes, we propose utilizing multiple repetitions of a periodic hyper-toroidal domain. We show that the naïve use of the ϕ p criterion for the construction of optimized designs can produce undesired orthogonal grid patterns (either complete or incomplete). We show how this behavior is related to the directional non-uniformity of hypercubical volume considered in the objective function, and we propose a simple remedy that involves limiting the interaction to a rotationally symmetrical neighborhood. Use of the recently proposed minimum image convention may provide too crude an approximation of the full periodic extension of the design space. We propose that a finite but sufficiently large interaction radius be considered for the evaluation of the pairwise potential. The upper bound on the interaction radius can be set to contain a sufficient number of points within the periodically repeated domain. These enhancements are embodied in the proposed ψ p criterion for space-filling designs. We show that the new criterion favors designs with better space-filling property, better projection properties and also with lower discrepancy. Euclidean distances among points within high-dimensional objects tend to concentrate and the resolution between distances decreases. We show that despite the decreasing contrast of distances, the desired resolution ability of the refined criterion is retained even when this isotropic metric is used. [ABSTRACT FROM AUTHOR]

• Mesoscale discrete model of concrete enhanced by spatially random material parameters. • Notched and unnotched three point bending and uniaxial tension are analyzed. • Fracture process zone statistically evaluated for various random field parameters. • Nontrivial dependence of peak loads on correlation length and variance is reported. • Local averaging and weakest-link are identified as the underlying mechanisms. Numerical simulations of concrete fracture performed with a probabilistic mesoscale discrete model are presented. The model represents a substantial part of material randomness by assigning random locations to the largest aggregates. The remaining part of randomness is introduced by causing material parameters to fluctuate randomly via a homogeneous random field. An extensive numerical study performed with the model considers prisms loaded in uniaxial tension with both fixed and rotating platens, and also beams with and without a notch loaded in three point bending. The results show the nontrivial effect of (i) autocorrelation length and (ii) variance of the random field on the fracture behavior of the model. Statistics of the peak load are presented as well as the size and shape of the fracture process zone at the moment when the maximum load is attained. Local averaging within the fracture process zone and weakest-link are identified as underlying mechanisms explaining the reported results. The companion paper, Part II (Vořechovský and Eliáš, 2020), introduces an analytical model capable of predicting the distribution of the peak load obtained with the probabilistic discrete model via the simple estimation of extremes of a random field obtained as moving average of local strength. [ABSTRACT FROM AUTHOR]

• Strength prediction of quasibrittle material that removes problems of Weibull theory. • Statistics of peak load are obtained analytically based on extremes of local averages. • Local averaging of Gaussian random field provides effective strength of material. • The weakest-link mechanics is incorporated via extremes of effective strength field. • Four model parameters are used to represent a wide range of random field parameters. The paper delivers an analytical model for prediction of the peak force in concrete specimens loaded in bending (both notched and unnotched). The model is capable of predicting peak force statistics by computing the extreme values of sliding averages of random strength fields. The local strength of the specimen is modeled by a stationary isotropic random field with Gaussian distribution and a given autocorrelation function. The averaging operation represents the progressive loss in material integrity and the associated stress redistribution that takes place prior to reaching the peak load. Once the (linear) averaging process is performed analytically, the resulting random field of averaged strength is assumed to represent a series of representative volume elements (RVEs) and the global strength is found by solving for the minimum of such an effective strength field. All these operations can be written analytically and there are only four model parameters: the three dimensions of the averaging volume (RVE) and the length of the final weakest-link chain. The model is verified using detailed numerical computations of notched and unnotched concrete beams simulated by mesoscale discrete simulations of concrete fracture performed with probabilistic distributions of model parameters. These are presented in the companion paper Part I (Eliáš and Vořechovský, 2020). The numerical model used for verification represents material randomness both by assigning random locations to the largest aggregates and by simulating random fluctuations of material parameters via a homogeneous random field. [ABSTRACT FROM AUTHOR]

• Uniform distribution of points in a unit hypercube. • Optimal Design of Experiments for computationally intensive models. • The analogy to a dynamical system of repulsive particles in periodic space is exploited. A method is proposed for the construction of uniformly distributed point sets within a design domain using an analogy to a dynamical system of interacting particles. The possibility of viewing various distance-based optimality criteria as formulas representing the potential energy of a system of charged particles is discussed. The potential energy is employed in deriving the equations of motion of the particles. The particles are either attracted or repelled and dissipative dynamical systems can be simulated to achieve optimal and near-optimal arrangements of points. The design domain is set up as an N var -dimensional unit hypercube, with N var being the number of variables (factors). The number of points is equal to the number of simulations (levels). The periodicity assumption of the design domain is shown to be an elegant way to obtain statistically uniform coverage of the design domain. The ϕ p criterion, which is a generalization of the Maximin criterion, is selected in order to demonstrate its analogy with an N-body system. This criterion guarantees that the points are spread uniformly within the design domain. The solution to such an N-body system is presented. The obtained designs are shown to outperform the existing optimal designs in various types of applications: multidimensional numerical integration, statistical exploration of computer models, reliability analyses of engineering systems, and screenings or exploratory designs for the global optimization/minimization of functions. [ABSTRACT FROM AUTHOR]

The paper deals with the classical fiber bundle model with brittle Weibull fibers and equal load sharing. This model is significant for the strength of fibrous materials and composites, and the generally random strength of quasi-brittle structures. The model is also relevant for the analysis of the reliability of various parallel systems (computer components, infrastructure etc.). An improved closed-form Gaussian approximation of the standard deviation of bundle strength is presented and compared with accurate solution calculated by recursive formula proposed by Daniels. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

The article introduces the BEE-ST approach for precise spatiotemporal monitoring of tooth and bone development, revealing novel insights. Topics include quantification of tooth turnover rates, dietary effects on molar root growth, and dynamics of bone and tooth growth across various species; and the tool holds promise for developmental biology, tissue engineering, and disease modeling.

Abstract: The article proposes an improvement over the widely used sequentially linear solution procedure often utilized for fracture simulations. In the classical secant version of this method, a partial solution of a step is scaled to reach a stress limit in exactly one element and the mechanical properties of the critical element are reduced. Non-proportional loading is generally unfeasible due to avalanches of ruptures caused by stress redistribution. Because only one loading vector can be scaled at a time, all others have to remain constant during the step. However, the constant load vectors do not allow proper determination of the critical element. A modified procedure based on redistribution of released stresses is developed here. It preserves the linearity of each step. After rupture of the critical element, a sequentially linear redistribution process of stress release takes place until a static equilibrium state is reached. During the redistribution, other elements may break. The proposed enhanced sequential procedure is also compared with another recently published “event-by-event” linear method for non-proportional loading. It is shown here, with the help of simple examples, that the proposed redistribution method yields correct results for non-proportional loading, unlike the other methods under comparison. [Copyright &y& Elsevier]

• Crack tracing algorithm can be combined with a variety of crack-bridge models. • Cracks are implicitly detected on a one-by-one basis. • Implementation using open-source scientific computing tools with a web application. The paper describes a probabilistic model capturing the multiple-cracking behavior of unidirectional brittle-matrix composites loaded in tension. The approach to modeling of composite fragmentation introduces two salient features that enhance both efficiency and flexibility compared to existing simulation methods: First, the algorithm identifies the emerging cracks one by one within a minimum number of load increments. Second, the crack-tracing algorithm is based on an abstract description of a crack bridge behavior. As a result, it is possible to combine the crack-tracing algorithm with a wide variety of crack bridge models, i.e. non-linear, deterministic or probabilistic. In this way, specific phenomena of bond behavior in different types of composites can be accounted for. The random nature of matrix cracking is reproduced using a random field simulation of the matrix strength. The model is verified by reproducing analytical results available for constant bond-slip law. The feasibility and robustness of the algorithm is demonstrated using an interactive web application that is directly executable from within a public github.com repository. [ABSTRACT FROM AUTHOR]

The paper presents a novel methodology to build surrogate models of complicated functions by an active learning-based sequential decomposition of the input random space and construction of localized polynomial chaos expansions, referred to as domain adaptive localized polynomial chaos expansion (DAL-PCE). The approach utilizes sequential decomposition of the input random space into smaller sub-domains approximated by low-order polynomial expansions. This allows the approximation of functions with strong nonlinearities, discontinuities, and/or singularities that often appear in dynamical systems. Decomposition of the input random space and local approximations alleviates the Gibbs phenomenon for these types of problems and confines error to a very small vicinity near the non-linearity. The global behavior of the surrogate model is therefore significantly better than existing methods, as shown in numerical examples, including an engineering dynamical system exhibiting discontinuous response. The whole process is driven by an active learning routine that uses the recently proposed Θ criterion to assess local variance contributions (Novák et al., 2021). The proposed approach balances both exploitation of the surrogate model and exploration of the input random space and thus leads to efficient and accurate approximation of the original mathematical model. The numerical results show the superiority of the DAL-PCE in comparison to (i) a single global polynomial chaos expansion and (ii) the recently proposed stochastic spectral embedding (SSE) method (Marelli et al., 2021) developed as an accurate surrogate model and which is based on a similar domain decomposition process. This method represents a general framework upon which further extensions and refinements can be based and which can be combined with any technique for non-intrusive polynomial chaos expansion construction. • Effective construction of a general approximation based on polynomial chaos expansion. • Novel method for domain decomposition and construction of local approximations. • Sequential domain decomposition and sample size extension based on an active learning. • Active learning represented by variance-based criterion for polynomial chaos expansion. [ABSTRACT FROM AUTHOR]