Static Condensation Research Articles

Periodic free vibrations of composite laminates with curvilinear fibres and CNTs

This article addresses the combined effect of using curvilinear fibres and carbon nanotubes (CNTs) reinforcements in the non-linear modes of vibration of laminated composite plates. To arrive at the material properties of the three-phase composite material, a two-step hierarchic procedure is followed. A modified version of the Halpin–Tsai model is employed to predict the Young’s modulus of the CNT enriched resin and expressions, deduced from equilibria of a unit cell where a fibre is embedded in resin, are applied to obtain the diverse elasticity moduli of the three-phase composite. Moderately large displacements are considered, with von Kármán strain–displacement relations. Although the presented model is an equivalent single layer one, it applies to thick plates, because a Third-order Shear Deformation Theory (TSDT) is followed. The set of autonomous non-linear equations of motion is reduced using static condensation and a modal basis with selected modes, chosen after a convergence analysis. The reduced set of equations of motion is solved by the shooting method. Numerical tests considering plates with diverse curvilinear fibre paths, CNT contents and thicknesses are carried out. The results obtained are thoroughly analysed.

A two-level semi-hybrid-mixed model for Stokes–Brinkman flows with divergence-compatible velocity–pressure elements

A two-level version for a recent semi-hybrid-mixed finite element approach for modeling Stokes and Brinkman flows is proposed. In the context of a domain decomposition of the flow region Ω, composite divergence-compatible finite elements pairs in H(div,Ω)×L2(Ω) are utilized for discretizing velocity and pressure fields, using the same approach previously adopted for two-level mixed Darcy and stress mixed elasticity models. The two-level finite element pairs of spaces in the subregions may have richer internal resolution than the boundary normal trace. Hybridization occurs by the introduction of an unknown (traction) defined over element boundaries, playing the role of a Lagrange multiplier to weakly enforce tangential velocity continuity and Dirichlet boundary condition. The well-posedness of the method requires a proper choice of the finite element space for the traction multiplier, which can be achieved after a proper velocity FE space enrichment with higher order bubble fields. The method is strongly locally conservative, yielding exact divergence-free velocity fields, demonstrating pressure robustness, and facilitating parallel implementations by limiting the communication of local common data to at most two elements. Easier coupling strategies of finite elements regarding different polynomial degree or mesh widths are permitted, provided that mild mesh and normal trace consistency properties are satisfied. Significant improvement in computational performance is achieved by the application of static condensation, where the global system is solved for coarse primary variables. The coarse primary variables are a piecewise constant pressure variable over the subregions, velocity normal trace and tangential traction over subdomain interfaces, as well as a real number used as a multiplier ensuring global zero-mean pressure. Refined details of the solutions are represented by secondary variables, which are post-processed by local solvers. Numerical results are presented for the verification of convergence histories of the method.

Mixed variational formulation and finite-element implementation of second-order poro-elasticity

We present a variational formulation of second-order poro-elasticity that can be readily implemented into finite-element codes by using standard Lagrangian interpolation functions. Point of departure is a two-field minimization principle in terms of the displacement and the fluid flux as independent variables. That principle is taken as a basis for the derivation of continuous and incremental saddle-point formulations in terms of an extended set of independent variables. By static condensation this formulation is then reduced to a minimization principle in terms of the displacement and fluid flux as well as associated higher-order fields. Once implemented into a finite-element code, the resulting formulation can be applied to the numerical simulation of porous media in consideration of second-order effects. Here, we analyze the model response by means of several example problems including two standard tests in poro-elasticity, namely the consolidation problems of Terzaghi and Mandel, and compare the results with those of a corresponding first-order model. As becomes clear, the second-order formulation can unleash its full potential when applied to the study of porous media having spatial dimensions comparable to the size of their microstructure. In particular, it is capable to regularize steep field gradients at external as well as internal surfaces and to describe material dilatation effects known from experiments.

A multiscale steel–concrete interface model for structural applications

In this paper, a new multiscale macro-element formulation for the steel–concrete interface modeling is proposed. This element allows for the representation of the behavior of steel and the interface zone surrounding it. It can also model the interfacial bond stresses in between. Compared to conventional interface models of the literature, which employ separate mesh elements for the steel and the interface, utilizing a macro-element to model both components simplifies the creation of a reinforced concrete structure mesh. Moreover, the macro-element equilibrium is solved using a sub-structuring method that aims to reduce the computational cost. At the global level, it is considered as a four-node element linked to two-dimensional and three-dimensional concrete elements. At the local level, an assembly of multiple three-node bar elements with bond stresses is performed. An inner mesh discretization is therefore possible at the local level independently of the global level. The coupling between the two modeling scales is done using a static condensation technique. The formulation of the macro-element is presented in this paper. A selection of numerical examples is provided. The presented applications demonstrate the robustness of the proposed interface model and its capacity to reproduce the experimental behavior of reinforced concrete structural elements.

Improvement to dynamic condensation of structural systems using an innovative virtual model reduction

Model reduction methods have primarily been used in the field of the structural dynamic analysis to reduce the computational time. However, it has progressively been extending to other engineering practices like the damage detection to extract the structural response by means of limited monitored degrees-of-freedom. This paper aims to improve the iterative dynamic condensation method, as a convenient method of the model reduction, for better assessment of the structural dynamic properties. The improvement consists of: (1) Capturing different modal frequencies by applying an iterative perturbation near the frequency obtained from the static condensation. The perturbation is evaluated in a manner that there is no need to solve eigenvalue problem successively; (2) Developing virtual dynamic condensation (VDC) procedure to generalize unique reduced stiffness and mass matrices, reflecting the influence of all modes of vibration, needed for a complete dynamic analysis. Finally, the accuracy of the proposed VDC method was proved by solving three examples.

Hybridized formulations of flux reconstruction schemes for advection-diffusion problems

We present the hybridization of flux reconstruction methods for advection-diffusion problems. Hybridization introduces a new variable into the problem so that it can be reduced via static condensation. This allows the solution of implicit discretizations to be done more efficiently. We derive an energy statement from a stability analysis considering a range of correction functions on hybridized and embedded flux reconstruction schemes. Then, we establish connections to standard formulations. We devise a post-processing scheme that leverages existing flux reconstruction operators to enhance accuracy for diffusion-dominated problems. Results show that the implicit convergence of these methods for advection-diffusion problems can result in performance benefits of over an order of magnitude. In addition, we observe that the superconvergence property of hybridized methods can be extended to the family of FR schemes for a range of correction functions.

Constraints of local stiffness adaption for reduced-order models of large-scale steel bridges

Large and complex structures, such as bridges, are subjected to a variety of environmental factors and loads over time, leading to evolving structural states. To assess their current condition and plan predictive maintenance, Digital Twins enriched with Structural Health Monitoring (SHM) data are essential. However, for large-scale structures, the creation of physics-based Digital Twins necessitates reduced-order modeling to ensure efficient adaptation to SHM data. Many of these methods for structural damage assessment rely on localized stiffness reduction to characterize discrete damage states. The Static Condensation Reduced Basis Element (SCRBE) method is a promising approach for model order reduction by decomposing the structure into parametric components. However, SCRBE's effectiveness and the accuracy of the Digital Twin depend on an appropriate partitioning of the structure into to components with reduced stiffness. This paper systematically investigates the process and the constraints of component selection for steel bridges with discrete damages, considering factors like numerics, sensor placement and complex geometries. A comprehensive analysis is conducted using a real bridge case study augmented with synthetic measurement data representing damage scenarios. The results reveal both challenges and the potential of the SCRBE method for Digital Twins. A systematic approach for component selection within the SCRBE framework is proposed, specifically tailored for real-world application scenarios. This structured methodology facilitates the efficient and adaptive development of Digital Twins for large, intricate structures, such as steel bridges.

Lego-like composite laminate construction and analysis on the fly

High-fidelity finite element (FE) analysis is computational demanding for parametric studies and design optimizations of composite laminates because it is time-consuming and inflexible in reconfiguration. Hence, we aim to develop rapid yet accurate 3-D FE analysis that can readily change (i) layer materials including fiber angles, (ii) the number of layers, and (iii) the stacking sequence of layers. To achieve these three features, we propose an offline-online framework hinging on static condensation (SC) integrated with port reduction (PR) and reduced basis element (RBE) methods, resulting in port-reduced static condensation (PR-SC) and port-reduced static condensation reduced basis element (PR-SCRBE) analyses. Thanks to the PR-SC and PR-SCRBE techniques, we may easily build a composite laminate using a variety of layer materials and any desired number of layers, akin to assembling Lego bricks, and swiftly analyze the constructed composite laminate. For demonstration, we assess both PR-SC and PR-SCRBE approximations using three different composite laminates with varying numbers of layers for each of three-point bending and tensile tests. Compared with FE analysis, PR-SC analysis exhibits insignificant accuracy loss in displacement and stress evaluation while obtaining meaningful computational speedup. In contrast, PR-SCRBE analysis shows moderate accuracy loss in displacement and stress evaluation while gaining considerable computational speedup. In conclusion, we recommend employing PR-SC for accuracy and PR-SCRBE for efficiency as per application requirements. The proposed two-stage framework may benefit many-query and real-time applications involving composite laminate analysis, including design optimization and virtual testing.

Component-based wing structure reconfiguration and analysis on the fly

To facilitate the structural design of an aircraft wing, we propose a simulation technique that can not only readily reshape a wing structure, but also rapidly yet accurately analyze its responses. For swift wing reconfiguration, we express a wing geometry with respect to shape parameters, such as a taper ratio, a half wingspan, and a sweep angle, by geometric parameterization. For rapid yet accurate analysis, we simulate a large-sized wing structure using small-sized common components whose solution spaces are shrunken by port-reduced static condensation reduced basis element (PR-SCRBE) approximation. As a result, we can easily modify both the shape and topology of a wing structure in the context of repetitive analysis. For demonstration, we construct six exemplary wing structures by assembling deformed duplicates of five common wing components. With the six wing structures, we examine the accuracy and efficiency of linear elastostatic PR-SCRBE analyses in predicting displacements and stresses compared with the corresponding finite element (FE) analyses. Regarding accuracy, the six PR-SCRBE analyses yield displacements and stresses with average relative errors of less than 0.01% and 2.79%, respectively. As for efficiency, the six PR-SCRBE analyses evaluate displacements and stresses with average speedups of 12.58 and 11.56, respectively. In conclusion, the proposed approach for wing reconfiguration and analysis may produce rapid yet accurate structural predictions, thereby considerably assisting the structural design of an aircraft wing.

Improved implementation of concentrated plasticity models in real-time hybrid simulation

Real-time hybrid simulation (RTHS) is a cost-effective experimental testing technique for evaluating the dynamic response of structural systems. In RTHS, a portion of the structure is modeled numerically (i.e., numerical substructure), while the remaining part is tested physically (i.e., experimental substructure). These substructures are coupled in real time. A significant challenge in RTHS is accurately computing the dynamic response of the numerical substructure in real time during testing. Constrained by this challenge, numerical substructures in RTHS are often oversimplified (e.g., spring-mass-dashpot models), failing to capture more complex dynamic behaviors and limiting the applicability of RTHS. However, with the increasing use of innovative structural systems, new materials, and advanced design guidelines to ensure resilient and sustainable civil infrastructure against natural hazards, the demand for real-time executable higher-fidelity numerical substructures in RTHS is growing. Consequently, concentrated plasticity model (CPM) has gained attention as the numerical substructure in RTHS for its good balance between numerical accuracy and computational efficiency, as well as its popularity in structural engineering applications. Currently, static condensation is utilized to prevent an increase in computational degrees of freedom (DOFs) induced by the hinges in CPM, aiming to save computational cost. However, this method proves counterproductive for nonlinear analyses. While reducing the number of computational DOFs, it complicates the stiffness representation, thereby necessitating more demanding arithmetic operations for stiffness updating. In this study, multi-point constraints are applied to enhance the computational efficiency, specifically for nonlinear analyses involving stiffness updating, and the versatility of CPM. Moreover, direct stiffness updating method and sparse matrix-vector multiplication are introduced for further enhancement in computation. An open-source MATLAB/Simulink-based computational tool, namely the Hybrid2D, is developed to promote the use of the improved CPMs in RTHS as numerical substructures. Numerical simulations are first conducted to verify the accuracy and the computational efficiency of the improved CPMs in Hybrid2D. Subsequently, two RTHS examples are experimentally conducted to demonstrate the effectiveness of the Hybrid2D in enabling RTHS with a higher-fidelity numerical substructure for more complex structural systems.

Generalized polynomial chaos expansion by reanalysis using static condensation based on substructuring

Generalized polynomial chaos expansion by reanalysis using static condensation based on substructuring

A numerical investigation on coupling of conforming and hybridizable interior penalty discontinuous Galerkin methods for fractured groundwater flow problems

The present paper focuses on the numerical modeling of groundwater flows in fractured porous media using the codimensional model description. Therefore, fractures are defined explicitly as a (d−1)-dimensional geometric object immersed in a d-dimensional region and can act arbitrarily as a drain or a barrier. We numerically investigate a novel numerical strategy combining distinctive classes of conforming and nonconforming high-order Galerkin methods, both eligible for static condensation. This procedure is here particularly relevant, leading to a smaller and sparser final system with coupled degrees of freedom solely on the mesh skeleton. Precisely, we combine an inspired hybridizable interior penalty discontinuous Galerkin (HIP) formulation inside the bulk region and a standard continuous Galerkin (CG) approximation on the fracture network. The distinctive discretization of corresponding PDEs and the coupling strategy are rigorously exposed, and the local and global matrix assemblies are detailed. Extensive numerical experiments are then achieved to prove the model’s performances for 2D/3D analytic and realistic benchmarks. Qualitative comparisons are also considered with other discretization methods and commercial software, such as comsol.

An efficient shear and bending‐locking‐free quadrilateral plate element using a modified Hellinger‐Reissner functional and the Bergan free formulation

AbstractThis paper presents a general variational principle as theoretical support for creating a simple and efficient quadrilateral plate finite element called BKWA, having only a single displacement and two rotations at corner nodes according to the Reissner‐Mindlin plate theory. The functional is a modified Hellinger‐Reissner in terms of the kinematic variables and independent transverse shear strains. Hence, we consider quadratic rotations, as in DKQ, and linear independent shear strains with constant tangential components on sides, as in MITC4. The quadratic rotation contributions and the mixed shear strain components are wiped out by static condensation. To adequately satisfy the constant curvature patch tests, we also consider the orthogonality conditions between the linear and quadratic bending modes, as in the Bergan free formulation, to avoid ‘spurious bending energy’ for thick plates (also called bending locking). The paper contains comprehensive patch tests and convergence tests with regular and irregular meshes (displacements, internal forces, and strain energy) of circular and square plates. The element is compared with other existing elements (MITC4, DKMQ). It is concluded that BKWA is an efficient element regarding several criteria.

Achieving Higher Levels of Crack Simulation with the Improved Adaptive Static Condensation Method

Accurately simulating concrete cracking in large-dimensional structures is a challenging task. To address this issue, an adaptive static condensation (ASC) method has been developed that has demonstrated effectiveness in localized nonlinearities. The ASC method aims to concentrate computational efforts solely on the damaged area, which may evolve due to crack initiation or propagation. However, the efficiency of the ASC method may be limited as it is based on a non-evolving mesh. To overcome this limitation, a novel approach is proposed in this study, which utilizes an evolutionary mesh with mesh refinement. The proposed approach employs a fine mesh solely in the activated and evolving domain of interest. The ASC method with mesh refinement is demonstrated on a notched bending beam, indicating that the accuracy of the ASC is maintained while providing an additional gain in computational time. Furthermore, a reinforced concrete vessel subjected to internal pressure is considered, and it is shown that this new approach results in a significant improvement in computational time, with a 14-fold improvement compared to a 5-fold improvement without mesh refinement. This study demonstrates that the proposed improvement on the ASC method allows for finer discretization in the zones of interest that were previously inaccessible with the nominal ASC method or a direct numerical simulation strategy.

Complexity of direct and iterative solvers on space–time formulations and time-marching schemes for h-refined grids towards singularities

We study computational complexity aspects for Finite Element formulations considering hypercubic space–time full and time-marching discretization schemes for h-refined grids towards singularities. We perform a relatively comprehensive study comparing the computational time via time complexities of direct and iterative solvers. We focus on the space–time formulation with refined computational grids and the corresponding time slabs, namely, computational grids obtained by taking the “cross-sections” of the refined space–time mesh. We estimate the computational complexity of solving systems of linear equations for multidimensional meshes with arbitrary dimensional singularities encountered in space–time formulation and time-marching schemes. The choice between space–time formulation and time-marching schemes depends entirely on the problem’s nature and the properties that need to be addressed. Thus, the paper aims to discuss the computational complexities of both approaches rather than suggest a better formulation. Our considerations concern the computational complexity of sequental execution of the multi-frontal solvers, the iterative solvers, and the static condensation. Numerical experiments with Octave confirm our theoretical findings.

Incompatible-mode geometrically non-linear finite element for micropolar elasticity

In this work a new three-dimensional geometrically non-linear hexahedral micropolar finite element enhanced with incompatible modes is presented. The analytical model is expressed in terms of Biot-like stress and couple-stress tensors and corresponding Biot-like strain and curvature tensors, with a linear, elastic and isotropic constitutive law. The numerical model is derived based on the principle of virtual work, and the residual derivation together with the linearisation and static condensation procedure is given in detail. The newly developed finite element is tested against the analytical solution of the geometrically non-linear micropolar pure bending problem and the element accuracy and robustness is compared against hexahedral Lagrangian finite elements of first and second order on several numerical examples. It is shown that the newly presented element is fast convergent, more robust and more accurate than the available Lagrangian elements. Moreover, the operator split and static condensation provide for a significantly lower computational cost than standard elements.

Revising the Boundary Element Method for Thermoviscous Acoustics: An Iterative Approach via Schur Complement

The Helmholtz equation is a reliable model for acoustics in inviscid fluids. Real fluids, however, experience viscous and thermal dissipation that impact the sound propagation dynamics. The viscothermal losses primarily arise in the boundary region between the fluid and solid, the acoustic boundary layers. To preserve model accuracy for structures housing acoustic cavities of comparable size to the boundary layer thickness, meticulous consideration of these losses is essential. Recent research efforts aim to integrate viscothermal effects into acoustic boundary element methods (BEM). While the reduced discretization of BEM is advantageous over finite element methods, it results in fully populated system matrices whose conditioning deteriorates when extended with additional degrees of freedom to account for viscothermal dissipation. Solving such a linear system of equations becomes prohibitively expensive for large-scale applications, as only direct solvers can be used. This work proposes a revised formulation for the viscothermal BEM employing the Schur complement and a change of basis for the boundary coupling. We demonstrate that static condensation significantly improves the conditioning of the coupled problem. When paired with an iterative solution scheme, the approach lowers the algorithmic complexity and thus reduces the computational costs in terms of runtime and storage requirements. The results demonstrate the favorable performance of the new method, indicating its usability for applications of practical relevance in thermoviscous acoustics.

A novel view on the canonical ‘stretching-due-to-bending’ nonlinear effect: A slighly curved beam

The analysis of nonlinear coupling between flexural and longitudinal vibrations of ideally straight elastic beams is a classical topic in vibration theory. The boundary conditions usually applied in this analysis are formulated as immobile hinges, which generate the canonical ‘stretching-due-to-bending’ nonlinear effect. The key assumption in formulation of this model is neglecting the longitudinal inertia, which may be referred to as ‘static condensation’. The aims of this work are, first, to re-consider this problem from the viewpoint of a nonlinear theory of curved beams, and, second, employ alternative type of boundary conditions, known as class-consistent ones. A paradoxical difference in nonlinear parts of Duffing equations obtained in the limit of vanishing curvature of an initially curved beam and in the case of an ideally straight beam is demonstrated and explained.

Combining the hybrid mimetic mixed method with the Scharfetter-Gummel scheme for magnetised transport in plasmas

In this paper, we propose a numerical scheme for fluid models of magnetised plasmas. One important feature of the numerical scheme is that it should be able to handle the anisotropy induced by the magnetic field. In order to do so, we propose the use of the hybrid mimetic mixed (HMM) scheme for diffusion. This is combined with a hybridised variant of the Scharfetter-Gummel (SG) scheme for advection. The proposed hybrid scheme can be implemented very efficiently via static condensation. Numerical tests are then performed to show the applicability of the combined HMM-SG scheme, even for highly anisotropic magnetic fields.

A semi‐hybrid‐mixed method for Stokes–Brinkman–Darcy flows with H(div)‐velocity fields

SummaryWe consider new semi‐hybrid‐mixed finite element formulations for Stokes–Brinkman problems. Using ‐conforming approximate velocity fields, the continuity of normal components over element interfaces is taken for granted, and pressure is searched in discontinuous spaces preserving the divergence compatibility property. Tangential continuity is weakly imposed by a Lagrange multiplier playing the role of tangential traction. The method is strongly mass‐conservative, leading to exact divergence‐free simulations of incompressible flows. The Lagrange multiplier space requires specific choices according to the velocity approximations implemented in each element geometry. In certain cases, classic divergence‐compatible pairs adopted for Darcy flows may require divergence‐free bubble enrichment to enforce tangential continuity in some extent, avoiding any extra stabilization technique. Considerable improvement in computational performance is achieved by the application of static condensation: the global system is solved only for a piecewise constant pressure variable, velocity normal trace and tangential traction over interfaces. The remaining solution components are recovered by solving independent local Neumann problems in each element. Numerical results are presented for a set of standard test cases in the field of Stokes–Brinkman–Darcy flows with known analytical solutions. The main convergence properties of the method are verified in the whole range of parameters, from Stokes to Darcy limits, as well as for the combined Stokes–Darcy scenario. The robustness of the method is also demonstrated for more challenging test problems with complex non‐conforming meshes with local refinement patterns as well as for curved geometry.