Additive Decomposition Research Articles

From ductile damage to unilateral contact via a point-wise implicit discontinuity

Ductile damage models and cohesive laws incorporate the material plasticity entailing the growth of irrecoverable deformations even after complete failure. This unrealistic growth remains concealed until the unilateral effects arising from the crack closure emerge. We address this issue by proposing a new strategy to cope with the entire process of failure, from the very inception in the form of diffuse damage to the final stage, i.e. the emergence of sharp cracks. To this end, we introduce a new strain field, termed discontinuity strain, to the conventional additive strain decomposition to account for discontinuities in a continuous sense so that the standard principle of virtual work applies. We treat this strain field similar to a strong discontinuity, yet without introducing new kinematic variables and nonlinear boundary conditions. In this paper, we demonstrate the effectiveness of this new strategy at a simple ductile damage constitutive model. The model uses a scalar damage index to control the degradation process. The discontinuity strain field is injected into the strain decomposition if this damage index exceeds a certain threshold. The threshold corresponds to the limit at which the induced imperfections merge and form a discrete crack. With three-point bending tests under pure mode I and mixed-mode conditions, we demonstrate that this augmentation does not show the early crack closure artifact which is wrongly predicted by plastic damage formulations at load reversal. We also use the concrete damaged plasticity model provided in Abaqus commercial finite element program for our comparison. Lastly, a high-intensity low-cycle fatigue test demonstrates the unilateral effects resulting from the complete closure of the induced crack.

Traffic speed prediction using GARCH‐GRU hybrid model

AbstractTraffic speed prediction is an integral part of an intelligent transportation system (ITS) because an advanced knowledge of traffic speed can help taking proactive preventive steps to avoid impending problems and it can also help in trip planning. Traffic speed data comprises a time series that may be modelled using any statistical or machine learning technique. In most of the cases, however, the performance of such models is bottlenecked due to heteroskedasticity usually present in such datasets. ARCH/GARCH family of models are generally used to model variance in such data. This paper presents a novel technique, termed as GARCH‐GRU, based on additive decomposition that splits data into random (residual) and deterministic parts. Random part is normalized using rolling standard deviation. GARCH (1, 1) is used to predict conditional variance of the residual and the predicted variance is then used in the basic model equation along with normalized residual that mimic white noise as required by the model. The data other than residual is modelled using a GRU model. The approach is applied to two datasets corresponding to a downtown road and a motorway. For comparison, the same datasets are exposed to three classical techniques; seasonal ARIMA, CNN and GRU techniques. The results demonstrate that the GARCH‐GRU technique outperforms others for random data of downtown road but fails to handle dynamic variations present in the motorway data.

A coupled model of transport-reaction-mechanics with trapping, Part II: Large strain analysis

A coupled finite strain chemo-transport-mechanical formulation with trapping is here proposed to extend a previous work set in the realm of small strain theory in continuum mechanics. The theory is rooted in non-equilibrium rational thermodynamics. The kinematics is based on a multiplicative decomposition of the deformation gradient to account for swelling and shrinking, thermal, elastic and inelastic contributions. Mass balance laws and balance of linear and angular momentum, as well as the laws of thermodynamics for a convecting body, are directly formulated in their material description, after specifications of some standard transformation rules between current and reference configuration. Thermodynamic restrictions are identified based on the functional dependence of the referential Helmholtz free energy density, which is chosen as the thermodynamic potential, and further subjected to a constitutive additive decomposition. Constitutive prescriptions for the chemical potentials, referential heat and mass fluxes, chemical kinetics and the generalized heat equation lead to the establishment of the governing equations. The theoretical framework is complemented by numerical simulations, highlighting the potential of the proposed formulation in multi-physics applications.

Temporal tendency, seasonality and relationship with climatic factors of Crimean-Congo Hemorrhagic Fever cases (East of Turkey: 2012–2021)

Crimean-Congo Hemorrhagic Fever continues to be an important public health problem by expanding its borders. To evaluate the temporal trend, seasonality, and relationship with the climatic factors of Crimean-Congo Hemorrhagic Fever. Study data included cases treated in two different tertiary healthcare institutions between 2012 and 2021. The demographic characteristics of the cases and the dates of admission to the hospital were determined, and they were matched with the average of the measurements (temperature, cumulative precipitation, relative humidity, wind speed) of two different meteorology stations in the study area. By calculating the crude incidence rates, the trend in years was investigated. Estimates were created by removing the incidence rates, seasonality, and trend components using the additive decomposition technique. The temporal relationship between incidence rates and climatic factors was evaluated with the help of the Autoregressive Distributed Lag Bound Test. Toda Yamamoto test was used for causality verification. The mean age of the cases (n = 974) included in the study was 47.6 ± 17.7 years, and the majority (57.3%) were in the group above 45 years of age. 56.6% of the cases were male and there was a male predominance in all age groups. Incidence rates ranged from 5.5 to 23.1/100,000 over the ten-year period and there was a significant upward trend (R2 = 0.691, p = 0.003). Cases of Crimean-Congo Hemorrhagic Fever that started in March, peaked in July and ended in October, showed a clear seasonality. A cointegration relationship was observed between case incidence rates and air temperature, cumulative precipitation, and relative humidity (p < 0.05 for all). Climatic factors can only indirectly affect the occurrence of Crimean-Congo Hemorrhagic Fever cases. However, climatic conditions that become progressively more favorable for vector ticks lead to the spread of the disease. The control measures to be taken should be prepared by considering the changing climatic conditions and prioritizing the risk groups. There is a need for information and awareness-raising studies about climate change and the growing dangers associated with it, also outside of endemic regions.

The Seasonal Variations Analysis of Permanent GNSS Station Time Series in the Central-East of Europe

Observations from permanent GNSS stations are actively used for the research and monitoring of geodynamic processes. Today, with the use of modern scientific programs and IGS products, it is possible to determine GNSS station coordinates and velocities at the level of a few millimeters. However, the scientific community constantly faces the question of increasing the accuracy of coordinate definitions to obtain more reliable data in the study of geodynamic phenomena. One of the main sources of errors is systematic measurement errors. To date, the procedure for their removal is still incomplete and imperfect. Also, during the processing of long-term GNSS measurements, it was found that the coordinate time series, after the removal of trend effects, are also characterized by seasonal variations, mainly of annual and semi-annual periods. We estimated the daily coordinate time series of 10 permanent GNSS stations in the central-eastern part of Europe from 2001 to 2019 and calculated the seasonal variation coefficients for these stations. The average value of the coefficients for the annual cycle for the N, E, and H components is −0.7, −0.2, and −0.7 mm, and for the semi-annual cycle the average value is 0.3, 0.4, and −0.5 mm. The obtained coefficients are less than 1 mm, which is why it can be argued that there is no seasonal component in the coordinate time series or that it is so small that it is a problematic task to calculate it. This practical absence of a seasonal component in long-term time series of GNSS coordinates, in our opinion, is partly compensated by the use of modern models of mapping functions (such as VMF3) for zenith tropospheric delays instead of the empirical GMF. To test the obtained results, we calculated the coefficients of seasonal variations for the sub-network of GNSS stations included in the category of the best EPN stations—C0 and C1. The values of the coefficients for the stations of this network are also less than 1 mm, which confirms the previous statement about the absence of a seasonal component in the long-term time series of coordinates. We also checked the presence of seasonal changes in the time series using the well-known decomposition procedure, which showed that the seasonal component is not observed because the content does not exceed 10% for additive decomposition and 20% for multiplicative decomposition.

A lean additive frailty model: With an application to clustering of melanoma in Norwegian families.

Additive frailty models are used to model correlated survival data. However, the complexity of the models increases with cluster size to the extent that practical usage becomes increasingly challenging. We present a modification of the additive genetic gamma frailty (AGGF) model, the lean AGGF (L-AGGF) model, which alleviates some of these challenges by using a leaner additive decomposition of the frailty. The performances of the models were compared and evaluated in a simulation study. The L-AGGF model was used to analyze population-wide data on clustering of melanoma in 2391125 two-generational Norwegian families, 1960-2015. Using this model, we could analyze the complete data set, while the original model limited the analysis to a restricted data set (with cluster sizes ). We found a substantial clustering of melanoma in Norwegian families and large heterogeneity in melanoma risk across the population, where 52% of the frailty was attributed to the 10% of the population at highest unobserved risk. Due to the improved scalability, the L-AGGF model enables a wider range of analyses of population-wide data compared to the AGGF model. Moreover, the methods outlined here make it possible to perform these analyses in a computationally efficient manner.

Impact of COVID-19, the War in Ukraine, and Inflation on Interest in Museums in Poland

This article quantifies the factors that affect interest in the topic of museums in Poland from 2018 to 2022. The article analyzes the time series of web queries related to museums. For the years 2018–2019, an additive time series decomposition model is prepared to define the trend and seasonal variations. The results provide an inaccurate forecast for the remaining period, which indicates a change in already existing tendencies. Therefore, a comprehensive econometric model is created that includes indepen-dent variables describing the pandemic, the invasion of Ukraine, and inflation. The results of both parts of the study confirm that there is a growing long-term trend in interest in museums. The interest is cha-racterized by significant seasonality, favoring the summer period, the Night of Museums, and several bank holidays over Christmas and the winter period. The pandemic, the war in Ukraine, and inflation had a direct negative impact on the measured value. The former also contributed to the flattening of the growing trend. Although the latter two factors are statistically significant, their impact is not as high as in the case of the pandemic. Meanwhile, the seasonality pattern remains stable throughout the analysis period.

Mechanics and modeling of cold rolling of polymeric films at large strains — A rate-independent approach

Recently a new phenomenon of bonding polymeric films in solid-state, via symmetric rolling, at ambient temperatures (≈20 °C) well below the glass transition temperature (Tg≈78°C) of the polymer has been reported. In this new type of bonding, polymer films are subject to plane strain active bulk plastic compression between the rollers during deformation. Here, we analyze these plane strain cold rolling processes, at large strains but slow strain rates, by finite element modeling. We find that at low temperatures, slow strain rates, and moderate thickness reductions during rolling (at which the Bauschinger effect can be neglected for the particular class of polymeric films studied here), the task of material modeling is greatly simplified and enables us to deploy a computationally efficient, yet accurate, finite deformation rate-independent elastic–plastic material behavior (with the inclusion of isotropic-hardening). The finite deformation elastic–plastic material behavior based on (i) the additive decomposition of stretching tensor ( D = De+Dp, i.e., a hypoelastic formulation) with incrementally objective time integration and, (ii) multiplicative decomposition of deformation gradient (F=FeFp) into elastic and plastic parts, are programmed and carried out for cold rolling within Abaqus Explicit. Frictional interactions are modeled using a consistent rate-independent Coulombic law. Predictions from both formulations, i.e., hypoelastic and multiplicative decomposition, closely match the experimentally observed rolling loads. No specialized hyperelastic/visco-plastic material model is required to describe the behavior of the particular blend of polymeric films under the conditions described here, thereby significantly speeding up the computation for steady-state rolling simulations. It is revealed that under the deformation conditions when principal axes show negligible rotation, hypoelastic formulation can be valid at large elastic stretches. Moreover, the use of classical rigid-plastic modeling (often applicable to metals) is found to greatly underestimate the rolling loads for polymers due to large elastic stretches in the polymer films at large strains. Deformation aspects of solid-state polymeric sheets presented here are expected to facilitate the development of new processes involving (or related to) cold roll-bonding of polymers.

A variational level set model based on additive decomposition for segmenting noisy images with intensity inhomogeneity

The variational level set model is widely used in image segmentation. However, it suffers from the limited performance when the images are contaminated by noise and intensity inhomogeneity. To solve this problem, a variational level set model based on additive decomposition is proposed in this paper. First, we decompose the image into three components: structure, intensity inhomogeneity and noise, which are regularized by different metrics. Specially, the structure component is regularized as a piecewise constant function in bounded variation (BV) space; the intensity inhomogeneity is modeled as a smooth function that is regularized by H1 space, and the residual containing noise is measured by L2 norm. Furthermore, a new indirect regularization term for level set function is designed to enhance the accuracy of the segmentation outcomes. And then, alternate direction iteration algorithm combining with gradient descent and ADMM to solve the proposed model. Experiments on both synthetic and real images validate the proposed model. Compared with five state-of-the-art models, the experimental results demonstrate that the proposed model exhibits the significant improvement in both accuracy and efficiency, outperforming all other methods. Quantitative evaluations show the average segmentation coefficients for Jaccard, Dice and Accuracy against ground truth are 0.93, 0.95 and 0.96, respectively, and the average running time is 0.80 seconds, further confirming its superior performance.

Ambient ozone at a rural Central European site and its vertical concentration gradient close to the ground.

The representativeness of ambient air quality of an in situ measurement is key in the use and correct interpretation of the measured concentration values. Though the horizontal representativeness aspect is generally not neglected in air pollution studies, a detailed, high-resolution vertical distribution of ambient air pollutant concentrations is rarely addressed. The aim of this study is twofold: (i) to explore the vertical distribution of ground-level ozone (O3) concentrations measured at four heights above the ground-namely at 2, 8, 50, and 230m-and (ii) to examine in detail the vertical O3 concentration gradient in air columns between 2 and 8, 8 and 50, and 50 and 230m above the ground. We use the daily mean O3 concentrations measured continuously at the Košetice station, representing the rural Central European background ambient air quality observed during 2015-2021. We use the semiparametric GAM (generalised additive model) approach (with complexity or roughness-penalised splines implementation) to analyse the data with sufficient flexibility. Our models for both O3 concentrations and O3 gradients use (additive) decomposition into annual trend and seasonality (plus an overall intercept). The seasonal and year-to-year patterns of the modelled O3 concentrations look very similar at first glance. Nevertheless, a more detailed look through O3 gradients shows that they differ substantially with respect to their seasonal and long-term dynamics. The vertical O3 concentration gradient in 2-230m is not uniform but changes substantially with increasing height and shows by far the highest dynamics near the ground between 2 and 8m, differing in both the seasonal and annual aspects for all the air columns inspected. We speculate that non-linear changes of both seasonal and annual components of vertical O3 gradients are due to atmospheric-terrestrial interactions and to meteorological factors, which we will explore in a future study.

Multi-layer additive tensor decomposition of infrared video for titanium alloy tensile testing

Infrared video (in mathematics terms, tensor) has been widely used in the tensile testing of metallic materials, such as titanium alloy and steel. The infrared video of the tensile testing process can effectively and efficiently determine the properties of metallic materials, e.g., Young’s modulus, Poisson’s ratio, yield strength, etc. The infrared video with structural properties, such as smoothness and sparsity, can be used to characterize the tensile testing process. To extract the features in the infrared video with structural properties, we propose a multi-layer additive tensor decomposition (MLATD) method based on regularization tensor regression for tensile testing. It decomposes a tensor into three classes of components: the multi-smooth layers (including background and foreground), the sparse layers (including between-tensor and in-tensor), and the noise layer. The scree plot is proposed to determine the number of multi-smooth layers, which is a downward curve of the difference between the smooth layers and the sparse layer. The alternating direction method of multipliers (ADMM) algorithm is proposed to solve the proposed method. The decomposition results of the simulation data and real-world case study revealed that the proposed method outperforms the existing state-of-the-art methods.

Command-Filtered Backstepping Repetitive Control for a Class of Uncertain Nonlinear Systems Based on Additive State Decomposition

This paper presents a command-filtered backstepping repetitive control (RC) method for a class of uncertain nonlinear systems with multiple disturbances, which is based on additive state decomposition (ASD) technique and active disturbance rejection approach. First, the original nonlinear system is decomposed into a linear time-invariant (LTI) primary system with periodic inputs and a nonlinear secondary system with multiple, aperiodic disturbances. The original nonlinear tracking problem is accordingly decomposed into two subproblems: RC problem for an LTI periodic system and a robust stabilization problem for a nonlinear system. By making using of the known information of the model, a linear extended state observer is constructed to estimate the matched total disturbance together with the system states and then a command-filtered backstepping control law incorporating with the estimate of the equivalent input disturbance is developed for the secondary system that ensures the robust stability and compensates for the effects of the multiple disturbances and residual nonlinearity. Stability criterion for each subsystem is established. Moreover, a particle-swarm-optimization algorithm is adopted to simultaneously optimize the controller parameters of the system. Finally, simulations and comparative experiments demonstrate the validity and superiorities of this method.

A physics-informed deep neural network for surrogate modeling in classical elasto-plasticity

In this work, we present a deep neural network architecture that can efficiently surrogate classical elasto-plastic constitutive relations. The network is enriched with crucial physics aspects of classical elasto-plasticity, including additive decomposition of strains into elastic and plastic parts, and nonlinear incremental elasticity. This leads to a Physics-Informed Neural Network (PINN) surrogate model named here as Elasto-Plastic Neural Network (EPNN). Detailed analyses show that embedding these physics into the architecture of the neural network facilitates a more efficient training of the network with less training data, while also enhancing the extrapolation capability for loading regimes outside the training data. The architecture of EPNN is model and material-independent; it can be adapted to a wide range of elasto-plastic material types, including geomaterials; and experimental data can potentially be directly used in training the network. To demonstrate the robustness of the proposed architecture, we adapt its general framework to the elasto-plastic behavior of sands. We use synthetic data generated from material point simulations based on a relatively advanced dilatancy-based constitutive model for granular materials to train the neural network. The superiority of EPNN over regular neural network architectures is demonstrated through predicting unseen strain-controlled loading paths for sands with different initial densities.

A nonlinear constitutive model for whole stress–strain behaviors of compressed rocks incorporating crack closure effect

AbstractThis paper is devoted to establishing a nonlinear constitutive model incorporating crack closure effect for capturing the whole process of rock deformation and failure behavior under compression. The model formulated in the framework of irreversible thermodynamics is based on the additive decomposition of the total strain into crack closure strain, elastic, and plastic parts. New specific criteria are proposed for the description of crack closure strain and plastic strain evolution. Notably, analytical solutions of the model under conventional triaxial compression loading conditions are derived. For application, a robust semi‐implicit return mapping (SRM) algorithm with a crack closure strain correction and a plastic strain correction is developed. The analytical solutions, herein, are subtly used to calibrate model parameters and as reference results for assessing the accuracy and robustness of the SRM algorithm. Validation of the model conducted by comparing with experimental data from the literature shows that the model can properly describe the nonlinear mechanical behaviors of rocks, including crack closure effect, strain hardening/softening, peak and residual strength, as well as volumetric compaction/dilation transition. In addition, a nonlocal formulation is introduced as an extension of the model to remedy the mesh dependency problem.

Infrared spectra in amorphous alumina: A combined ab initio and experimental study

We present a combined study based on experimental measurements of infrared (IR) dielectric function and first-principles calculations of IR spectra and vibrational density of states (VDOS) of amorphous alumina (am-Al$_2$O$_3$). In particular, we show that the main features of the imaginary part of the dielectric function $\epsilon_2(\omega)$ at $\sim$380 and 630 cm$^{-1}$ are related to the motions of threefold coordinated oxygen atoms, which are the vast majority of oxygen atoms in am-Al$_2$O$_3$. Our analysis provides an alternative point of view with respect to an earlier suggested assignment of the vibrational modes, which relates them to the stretching and bending vibrational modes of AlO$_{n}$ ($n=$ 4, 5, and 6) polyhedra. Our assignment is based on the additive decomposition of the VDOS and $\epsilon_2(\omega)$ spectra, which shows that: (i) the band at $\sim$380 cm$^{-1}$ features oxygen motions occurring in a direction normal to the plane defined by the three nearest-neighbor aluminum atoms, i.e. out-of-plane motions of oxygen atoms; (ii) Al--O stretching vibrations (i.e. in-plane motions of oxygen atoms) appear at frequencies above $\sim$500 cm$^{-1}$, which characterize the vibrational modes underlying the band at $\sim$630 cm$^{-1}$. Aluminum and fourfold coordinated oxygen atoms contribute uniformly to the VDOS and $\epsilon_2(\omega)$ spectra in the frequency region $\sim$350--650 cm$^{-1}$ without causing specific features. Our numerical results are in good agreement with the previous and presently obtained experimental data on the IR dielectric function of am-Al$_2$O$_3$ films. Finally, we show that the IR spectrum can be modeled by assuming isotropic Born charges for aluminum atoms and fourfold coordinated oxygen atoms, while requiring the use of three parameters, defined in a local reference frame, for the anisotropic Born charges of threefold coordinated oxygen atoms.

Values of games over Boolean player sets

In this paper, we study new classes of value operators for coalitional games with players organized into a boolean algebra. Coalitional games are cooperative models in which players can form coalitions to maximize profit. The basic solution concepts in such game scenarios are value operators, which assign a unique real value to every player, reflecting thus selected principles of economic rationality. Some value concepts were extended beyond the classic coalitional model where every coalition of players can form. In particular, the extension of Shapley value exists for coalitional games in which players are partially ordered, and the feasible coalitions are the corresponding down-sets. Interestingly, this game-theoretic framework was employed in the method called Information Attribution. This method aims to solve the information decomposition problem, which asks for a particular additive decomposition of the mutual information between the input and target random variables. In such information-theoretic games, the players are predictors, and their set has the natural structure of a boolean algebra. Motivated by the original problem, we consider coalitional games where the players form a boolean algebra, and the coalitions are the corresponding down-sets. This more general approach enables us to study various value solution concepts in detail. Namely, we focus on the classes of values that can represent alternatives to the solution of the information decomposition problem, such as random-order values or sharing values. We extend the axiomatic characterization of some classes of values that were known only for the standard coalitional games.

Structural decomposition analysis of Brazilian greenhouse gas emissions

This paper aims to verify the long run structural changes in Greenhouse Gas (GHG) emissions in Brazil which presents the particularity of high emissions in the agriculture and livestock sectors, and low emissions in energy sectors. We use an additive Structural Decomposition Analysis (SDA) to verify the sectorial changes in emissions derived from the effects of economic intensity, final demand structure effect and technological change by component i.e., household consumption, exports, gross fixed capital formation, government consumption, inventory change and total final demand. The database is composed of a series of input-output matrices at constant prices for the period 2000–2017, with comparable production structure of 42 sectors. Our results show that agriculture had the most remarkable effects on emissions, followed by industry sectors for all SDA effects and components except for inventory change. Finally, we discuss relationships between energy, GHG emissions and production structure towards development policies to reduce emissions and improve energy use.

Multifidelity Neural Network Formulations for Prediction of Reactive Molecular Potential Energy Surfaces

This paper focuses on the development of multifidelity modeling approaches using neural network surrogates, where training data arising from multiple model forms and resolutions are integrated to predict high-fidelity response quantities of interest at lower cost. We focus on the context of quantum chemistry and the integration of information from multiple levels of theory. Important foundations include the use of symmetry function-based atomic energy vector constructions as feature vectors for representing structures across families of molecules and single-fidelity neural network training capabilities that learn the relationships needed to map feature vectors to potential energy predictions. These foundations are embedded within several multifidelity topologies that decompose the high-fidelity mapping into model-based components, including sequential formulations that admit a general nonlinear mapping across fidelities and discrepancy-based formulations that presume an additive decomposition. Methodologies are first explored and demonstrated on a pair of simple analytical test problems and then deployed for potential energy prediction for C5H5 using B2PLYP-D3/6-311++G(d,p) for high-fidelity simulation data and Hartree-Fock 6-31G for low-fidelity data. For the common case of limited access to high-fidelity data, our computational results demonstrate that multifidelity neural network potential energy surface constructions achieve roughly an order of magnitude improvement, either in terms of test error reduction for equivalent total simulation cost or reduction in total cost for equivalent error.

Hygro-Thermo-Mechanical Analysis of Brick Masonry Walls Subjected to Environmental Actions

Masonry walls comprise an important part of the building envelope and, thus, are exposed to environmental effects such as temperature and moisture variations. However, structural assessment usually neglects the influence of these hygro-thermal loads and assumes ideal conditions. This paper presents a hygro-thermo-mechanical model and its application to simulate the impact of temperature- and moisture-related phenomena on the structural behavior of masonry walls. A fully coupled heat and mass transfer model is presented and a 2D finite element model is prepared to simulate the behavior of a brick masonry wall under various hygro-thermal scenarios. Two different mortars are considered: namely, cement mortar and natural hydraulic lime mortar. The results are evaluated in terms of temperature and moisture content distribution across the wall thickness. The hygro-thermal model is further extended to incorporate mechanical effects through the total strain additive decomposition principle. It is shown that the hygro-thermo-mechanical response of the brick masonry wall is a complex 2D phenomenon. Moreover, the environmental loads change the natural stress distribution caused by gravitational loads alone. Finally, the wall with cement mortar develops higher levels of stress when compared to the one with lime mortar, due to the dissimilar hygro-thermal behavior between the constituent materials.

The ternary diagram of functional diversity

Abstract Among the many diversity indices in the ecologist toolbox, measures that can be partitioned into additive terms are particularly useful as the different components can be related to different ecological processes shaping community structure. In this paper, an additive diversity decomposition is proposed to partition the diversity structure of a given community into three complementary fractions: functional diversity, functional redundancy and species dominance. These three components sum up to one. Therefore, they can be used to portray the community structure in a ternary diagram. Since the identification of community‐level patterns is an essential step to investigate the main drivers of species coexistence, the ternary diagram of functional diversity can be used to relate different facets of diversity to community assembly processes more exhaustively than looking only at one index at a time. The value of the proposed diversity decomposition is demonstrated by the analysis of actual abundance data on plant assemblages sampled in grazed and ungrazed grasslands in Tuscany (Central Italy).