Static Condensation Research Articles

A coupled finite element formulation for the modeling of freezing processes within the Theory of Porous Media using a discontinuous approximation for the volume fraction ice

AbstractThis work presents a multi‐phase approach to model freezing processes within a coupled finite element formulation using the Theory of Porous Media (TPM), see also [1]. The focus of the study is to enhance the numerical stability for the volume fraction ice as is it highly sensitive to phase discontinuities. To address this issue, a discontinuous approximation is employed for the volume fraction ice while the remaining degrees of freedom are still continuously approximated. As a result, the corresponding degrees of freedom are independently solved for each element, as the element nodes are not interconnected with other elements. Academic examples are provided within a standard Galerkin procedure to highlight the optimization effect, comparing the results obtained with a continuous and discontinuous approximation for the volume fraction ice. Additionally, the implementation of the discontinuous approximation allows for the application of static condensation, resulting in reduced computation time.

Efficient micromagnetic–elastic simulations based on a perturbed Lagrangian function

Micromagnetic simulations require the numerically challenging preservation of the Euclidean norm during the whole simulation. This can be accomplished by applying a priori length preserving methods, renormalization algorithms, or penalization strategies. The latter one includes both the penalty method and the Lagrangian multiplier. The penalty method requires the definition of a penalty parameter during the initiation of the simulation which, depending on its size, can lead to an unsatisfied constraint or stiff and difficult to solve systems of equations. The Lagrange multiplier always penalizes in problem-dependent intensity, hence, an additional degree of freedom is added to the system of equations to the drawback of higher computational costs. This paper proposes a method that utilizes a perturbed Lagrangian multiplier and an element level static condensation to condensate the additional degree of freedom. This guarantees fast simulations, and no parameter fitting in advance. Suitable numerical examples are conducted to prove the workability of the outlined scheme and to highlight the efficiency compared to the non-condensed formulation.

Pressure-robust enriched Galerkin methods for the Stokes equations

In this paper, we present a pressure-robust enriched Galerkin (EG) scheme for solving the Stokes equations, which is an enhanced version of the EG scheme for the Stokes problem proposed in Yi et al. (2022). The pressure-robustness is achieved by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. An a priori error analysis proves that the velocity error is independent of the pressure and viscosity. We also propose and analyze a perturbed version of our pressure-robust EG method that allows for the elimination of the degrees of freedom corresponding to the discontinuous component of the velocity vector via static condensation. The resulting method can be viewed as a stabilized H1-conforming P1−P0 method. Further, we consider efficient block preconditioners whose performances are independent of the viscosity. The theoretical results are confirmed through various numerical experiments in two and three dimensions.

Parametric estimation of a stability factor for certified reduced basis methods via adaptive Gaussian process

The a posteriori error analysis of a reduced basis (RB) method may certify the accuracy of an RB solution without the truth finite element (FE) solution. However, the error theory necessitates a numerically expensive stability factor such as coercivity or an inf–sup factor. Consequently, we perforce resort to the successive constraint method (SCM) to achieve a stability lower bound for a rigorous albeit rapid RB analysis. As an alternative, we propose an adaptive Gaussian process (AGP) to produce a stability lower bound for the rigorous, rapid, and sharp estimation of an RB solution error. We utilize linear elastostatic and Helmholtz acoustic problems with geometric parameterization to compare the AGP with the SCM for a numerical investigation. The results show that the AGP significantly surpasses the SCM. This is because the a posteriori error estimation of an RB solution obtained with the AGP is sharper and faster than that obtained with the SCM by factors of at least 103 and 10, respectively. With regard to a rigorous error estimation, the AGP succeeds for the linear elastostatic problem but fails for the Helmholtz acoustic problem when an RB size is one at parameter values incurring only resonance. Otherwise, it manages to attain a rigorous error estimation. To conclude, the AGP is observed to be heuristically more practical than the SCM for a certified RB analysis. This research may considerably aid a static condensation RB element (scRBE) method because the technique numerously repeats the a posteriori error estimation of a reduced bubble function for the a posteriori error estimation of an scRBE solution.

A conforming auxiliary space preconditioner for the mass conserving stress-yielding method.

We are studying the efficient solution of the system of linear equations stemming from the mass conserving stress-yielding (MCS) discretization of the Stokes equations. We perform static condensation to arrive at a system for the pressure and velocity unknowns. An auxiliary space preconditioner for the positive definite velocity block makes use of efficient and scalable solvers for conforming Finite Element spaces of low order and is analyzed with emphasis placed on robustness in the polynomial degree of the discretization. Numerical experiments demonstrate the potential of this approach and the efficiency of the implementation.

Parametric stiffness modeling of a variable cross-section link for a collaborative robot

This paper describes how a parametric stiffness model of the variable cross-section link of a collaborative robot is obtained. The variable cross-section link module is divided into three parts: the front connection part, the rear connection part, and the main part. The static condensation method is used to model the stiffness of the connecting parts. The parametric stiffness modeling of the main part is carried out using the structural mechanics method. The stiffness models of the three parts are integrated into the parametric stiffness model of the entire link. The parametric stiffness model of the variable cross-section link is related to the structural dimensions of the link. The elastic deformation errors in all directions obtained from the stiffness model are within 5%. The method establishes the mapping relationship between the link’s structural dimensions and its stiffness model. Based on this, the structural optimization of a collaborative robot based on parametric stiffness model is made possible.

Geometrically exact beam theory with embedded strong discontinuities for the modeling of failure in structures. Part I: Formulation and finite element implementation

We introduce a three-dimensional geometrically nonlinear Reissner beam theory with embedded strong discontinuities for the modeling of failure in structures and discuss its finite element implementation. Existing embedded beam theories are geometrically linear or two-dimensional, motivating the need for the present work. We propose a geometrically nonlinear beam theory that accounts for cracks through displacement discontinuities in 3D. To represent the three modes of fracture inside an element, we enrich each displacement field with an incompatible mode parameter in the form of a jump, an additional degree of freedom. We then eliminate these nodeless degrees of freedom through static condensation and evaluate them within the framework of inelasticity by utilizing an operator split solution procedure. Seeing that the coupled strain-softening problem is non-convex, we present an alternating minimization (staggered) algorithm, thus retaining a positive definite stiffness matrix. Finally, the four-parameter representation by quaternions describes a three-dimensional finite rotation. We demonstrate a very satisfying and robust performance of these new finite elements in several numerical examples, including the fracture of random lattice structures with application to fibrous materials. We show that accounting for geometrical nonlinearity in the beam formulation is necessary for direct numerical simulations of fiber networks regardless of the density.

Equivalent plate dynamic modeling of space periodic truss structures

Antenna structures used in spaceflight always contain nonlinear joints distributed along their cross section; hence, an equivalent beam model (EBM) that uses a point to describe the cross section is not ideally suited to depicting connection nonlinearity in the structure. To address this difficulty, a novel orthotropic equivalent plate model (EPM) of a space periodic truss structure is proposed. Firstly, utilizing the Mindlin plate hypothesis, strain in the periodic truss element is expanded into a binary high-order Taylor series, and corrections are introduced to normal strain. Secondly, employing the principle of energy equivalence, the equivalent elastic and inertial matrices of the EPM are calculated for in-plane and out-of-plane vibration. Finally, the static condensation method is utilized to reduce the dimensions of the equivalent model to establish an orthotropic equivalent Mindlin plate dynamic model, and its inherent characteristics are analyzed. The simulations show that when in-plane shear strain is expanded to the third order, the EPM can more accurately represent torsional vibration of the periodic truss structure. By introducing normal strain correction coefficients in the longitudinal direction, lateral bending vibrations of the periodic truss structure can be simulated more precisely. Compared with the EBM, the EPM can more accurately depict mechanical characteristics along the cross section. The EPM broadens the scope of continuum equivalent modeling of periodic trusses, and is more useful in engineering practice. In addition, the frequencies and mode shapes of the analytical EPM are consistent with a model using the finite element method, which supports the accuracy and effectiveness of the EPM. The analytical EPM provides a convenient way to study the nonlinear connections and vibration control of antenna structures with multiple joints along their cross section.

A simple and efficient lagrange multiplier based mixed finite element for gradient damage

A novel finite element formulation for gradient-regularized damage models is presented which allows for the robust, efficient, and mesh-independent simulation of damage phenomena in engineering and biological materials. The paper presents a Lagrange multiplier based mixed finite element formulation for finite strains. Thereby, no numerical stabilization or penalty parameters are required. On the other hand, no additional degrees of freedom appear for the Lagrange multiplier which is achieved through a suitable FE-interpolation scheme allowing for static condensation. In contrast to competitive approaches from the literature with similar efficiency, the proposed formulation does not require cross-element information and thus, a straightforward implementation using standard element routine interfaces is enabled. Numerical tests show mesh-independent solutions, robustness of the solution procedure for states of severe damage and under cyclic loading conditions. It is demonstrated that the computing time of the gradient damage calculations exceeds the one of purely elastic computations only by an insignificant amount. Furthermore, an improved convergence behavior compared to alternative approaches is shown.

Exact Model Order Reduction for the Full-System Finite Element Solution of Thermal Elastohydrodynamic Lubrication Problems

The derivation of fast, reliable, and accurate modeling procedures for the solution of thermal elastohydrodynamic lubrication problems is a topic of significant interest in the Tribology community. In this paper, a novel model order reduction technique is introduced for the analysis of thermal elastohydrodynamic lubrication problems. The method uses static condensation to reduce the size of the linear elasticity part within the overall matrix system, followed by a splitting algorithm to avoid the burden of solving a semi-dense matrix system. The results reveal the exactness of the proposed methodology, which does not introduce any additional model-reduction approximations to the overall solution. They also reveal the reduction in computational times, which is in the order of 10–20% for line contacts, while it is in excess of 50% for circular contacts. The robustness of the proposed method is displayed by using it to model some relatively highly loaded contacts whose numerical solution is known to be rather challenging.

Guided identification of nonlinear reduced-order models via the incorporation of von Kármán beam theory

A reduction in the number of Degrees Of Freedom (DOFs) of a nonlinear structural model can significantly decrease the computational time of dynamic analyses, especially in problems where long analyses are required. A Reduced-Order Model (ROM) of a structural system is commonly generated using some combination of dimensionality reduction, such as substitution of a modal basis or static condensation of membrane DOFs. Perhaps the most common and widespread framework for developing low-order models of continuous systems is the Ritz method in conjunction with a smooth modal basis. The Ritz method requires access to the closed form governing differential equations, and it is difficult to implement for problems with complex boundary conditions. In recent decades, an alternative framework that instead constructs a low-order model from an existing Finite Element Model (FEM) has been developed. Such models are typically known as NonLinear Reduced-Order Models (NLROMs). Such NLROMs can be used to handle complex boundary conditions so long as the FEM can do the same, and they allow the analyst to start from an existing finite element package as opposed to directly working with the governing differential equations. The form of the NLROM restoring force function is assumed by the analyst prior to identifying the model, and the parameters of the function are identified to achieve a high-quality fit to the FEM load–displacement data. This research aims to examine the connection between classically derived models using the Ritz method and NLROMs in order to predict the coefficients of NLROMs and to develop simplifying assumptions about their structure. Specifically, the Condensed Von Kármán (CVK) beam equation is used as a basis for the study of nonlinear beam structures. The Ritz method is applied to the CVK beam equation, and the matrix form of the equilibrium equation is determined. The equivalence between the CVK coefficients and the NLROM coefficients is shown, and a simplified, tangent stiffness-based identification procedure is proposed to determine the coefficients of the CVK-based model. An arc-length continuation algorithm is used to predict the static load–displacement response of the models studied and to assess the accuracy of the ROMs. In the case of initially straight, post-buckled beams, the proposed identification procedure is shown to generate a model that generally outperforms a corresponding NLROM identified using a more traditional displacement-based procedure.

Two Stage SEREP Method for Condensation of Space Frame Structure for Dynamic Analysis

Analysis of frame structure can be done by the computer analysis, but there is requirement of simple and easy model which can give accurate result or with tolerable accuracy. The process of reducing the size of matrix is called condensation. It needs to be converted into system or model having less number of degrees of freedom as huge computational effort is required to analyze big models. Also, the models which requires analysis multiple times, for example, optimization of structural system, condensation is very useful. The single stiffness value can represent the stiffness of the whole story (elastic, rectangular frame with evenly distributed load), which give the responses accurately or tolerable accuracy. With this we can create a single lumped mass system of the frame structure which come very handy in further analysis of structure. For finding the single stiffness value, there are various approach like by using Approximate Story Stiffness Formulas, Static condensation, System Equivalent Reduction Expansion Process (SEREP) etc. Approximate Story Stiffness formulae don’t give very accurate result. To overcome its shortcoming Static Condensation and SEREP methods are used. Static Condensation gives better result but there is significant difference in the responses. Especially for space frame structures it is difficult to directly condense in to single degrees of freedom at each floor level. Hence, the present study proposes two stage SEREP method for condensation of spaceframe structure. Firstly, space frame is reduced to single degrees of freedom at each node, then it is further reduced to single degree of freedom at each floor. The proposed method gives very less (acceptable) error in comparison of natural frequencies.

Hybrid method for full-field response estimation using sparse measurement data based on inverse analysis and static condensation

In structural health monitoring, measuring the accurate and spatially dense response near critical locations of the structure can be advantageous to estimate damage to the structure. Due to several physical restrictions or limitations of the sensing method, it may not always be possible to generate reliable data at critical locations. In this study, a hybrid method is presented that makes use of the measured displacement data and finite element (FE) model of the structure to predict dense full-field response. The presented method can incorporate unknown boundary conditions and unknown body forces by applying correction/fictitious forces to match predicted and measured responses. Using static condensation followed by inverse analysis, these additional forces are found by setting up a least square problem. Due to the problem being ill-posed, L2-penalty is used to control the prediction error. Numerical simulation of a plate subjected to body force showed an accurate prediction of full-field response except for a few boundary locations. To handle this, the proposed method is used in conjunction with linear interpolation near boundary locations. The method is validated in a laboratory experiment for a plate with a notch having displacement measured using Digital Image Correlation (DIC). On comparing strains calculated using predicted displacements, FEM, and DIC, the predicted strains show better agreement with the FEM than DIC. This affirms that the proposed hybrid technique can be used at critical locations where DIC fails to provide reliable strain data.

Improvement of Linear Tetrahedral Element Performance by Using Substructuring Method

We apply here a class of substructuring method to improve the performance of linear tetrahedral element used in finite element analysis (FEA). The method is novel and relied on the construction of mesh inside a tetrahedron volume which behaves as an assembly of substructures. The corresponding stiffness matrix of the mesh is assembled using a static condensation procedure which is used further to obtain strain energy from a set of particular displacement vectors. This energy is the key to obtain a so-called energy ratio that will modify the stiffness matrix of a linear tetrahedral element. In the numerical tests, we show that the method can improve the performance of the tetrahedral element to approximate displacement and stress fields from the analytical solutions for cantilever and stress concentration problems, respectively.

Assessment of an Assumed Strain-based Quadrilateral Membrane Element

This paper describes the development of a simple quadrilateral strain-based element for plane stress and strain problems. This element has five nodes, four located at its corners and one at the center. Each of the four corner nodes had two essential external degrees of freedom (u, v), while the center node had three degrees of freedom (u, v, ɵ); the static condensation method was used for the internal node. This element was used for both linear and dynamic analysis. Its performance was assessed using a variety of membrane and axisymmetric analysis problems. The obtained results demonstrated the good performance and accuracy of the proposed element.

A new reduced model of damped moment frame for rapid optimization of viscous damper placement

AbstractA new model reduction method is presented for moment‐resisting frames with viscous dampers. The proposed method considers (1) the interactive effect of added damping between the horizontal‐vertical directions and (2) the direct effect with regard to the vertical direction different from the conventional static condensation method. When brace‐type dampers are treated, these effects cannot be neglected originally. The proposed method gives a better correspondence of the fundamental eigenmode with that of the non‐reduced moment frame than the static condensation method. In addition, the proposed reduced model can accurately evaluate the maximum interstory drifts of the non‐reduced moment frame. The proposed model reduction method is formulated in the frequency domain. The frequency dependency of the stiffness matrix and the damping matrix of the reduced model is investigated. The influences of the added damping matrix by viscous dampers on the natural circular frequencies, the damping ratios and the participation vectors are also investigated. Finally, it is demonstrated through a numerical example that the displacement‐based optimization of the damper allocation can be accurately and rapidly conducted by using the reduced models throughout the optimization process.

Composite finite elements on dynamic and buckling responses of composite beams with independent rotations

To better understand the structural dynamic response and integrity under complex engineering situations, displacement-based variational principles and corresponding composite finite elements (CFEs) are developed for the simulation of vibration and buckling behavior of partial-interaction composite beams. The accuracy of the composite FEs are validated against the analytical and numerical results in the literature where good agreement is always generated. More importantly, the effect of independent-rotation assumption is investigated in the simulation which is tested to have significant impact on the natural frequencies and buckling loads of composite structures with distinct components and generally lead closer results to the plane-stress FE results. The influence of the rotation of sub-structure on the relative slips between sub-structures is discussed. This explains the discrepancies observed in the literature. The construction of CFEs also considers the internal degrees of freedom (DoFs), which are then eliminated through static condensation, to alleviate the shear/curvature locking phenomenon. The convergence of the proposed FEs are tested, along with the capability of the present FEs in avoiding the locking phenomenon. The final explicit expressions of the elements are also offered and can be readily implemented into the commercial packages with high computational efficiency.

Hybrid High-Order and Weak Galerkin Methods for the Biharmonic Problem

We devise and analyze two hybrid high-order (HHO) methods for the numerical approximation of the biharmonic problem. The methods support polyhedral meshes, rely on the primal formulation of the problem, and deliver $O(h^{k+1})$ $H^2$-error estimates when using polynomials of order $k\ge0$ to approximate the normal derivative on the mesh (inter)faces. Both HHO methods hinge on a stabilization in the spirit of Lehrenfeld--Sch\"oberl for second-order PDEs. The cell unknowns are polynomials of order $(k+2)$ that can be eliminated locally by means of static condensation. The face unknowns approximating the trace of the solution on the mesh (inter)faces are polynomials of order $(k+1)$ in the first HHO method which is valid in dimension two and uses an original stabilization involving the canonical hybrid finite element, and they are of order $(k+2)$ for the second HHO method which is valid in arbitrary dimension and uses only $L^2$-orthogonal projections in the stabilization. A comparative discussion with the weak Galerkin methods from the literature is provided, highlighting the close connections and the improvements proposed herein. Additionally, we show how the two HHO methods can be combined with a Nitsche-like boundary-penalty technique to weakly enforce the boundary conditions. An originality in the devised Nitsche's technique is to avoid any penalty parameter that must be large enough. Finally, numerical results showcase the efficiency of the proposed methods and indicate that the HHO methods can generally outperform discontinuous Galerkin methods and even be competitive with $C^0$-interior penalty methods on triangular meshes.

A framework for physics-driven generation of feature data for strain-based damage detection in aerospace sandwich structures

In recent years, structural health monitoring has been increasingly applied to composite sandwich structures, as typically used in aerospace applications. In addition, machine learning approaches are increasingly popular for damage detection, localization and size estimation, due to their great advantages in pattern recognition and anomaly detection. However, a major disadvantage of machine learning techniques is that these algorithms generally require large amounts of realistic data. In general, these data are expensive or even impossible to obtain within a feasible time. In order to overcome this hindrance, this work introduces a computationally inexpensive framework for physics-driven feature generation of strain data for the training of ML-based SHM methods using sub-structuring and the concept of reanalysis. First, the global FE model is subdivided into a monitored part, i.e., a smaller submodel, and a global model. Second, the stiffness matrix of the submodel is extracted from the finite element software. Then, static condensation is performed to further reduce the computational effort. Afterwards, selected eigenvectors are derived in terms of displacements of master nodes and the corresponding strains are calculated. Finally, a statistically varied linear combination between the different characteristic eigenvector load cases is performed based on the superposition principle. This procedure enables the efficient generation of a large number of different physics-driven determined strain solutions for a subsequent training of a ML algorithms. The proposed framework is evaluated by means of a damage detection approach, based on an artificial neuronal network classifier algorithm. The applied approach utilizes strain measurements from selected positions as physical quantity and is demonstrated using a composite sandwich structure imitating an aircraft spoiler. The key principle of the damage detection algorithm is based on the fact that a change in the relationship between sensors indicates the presence of damage. Additionally, to the numerical healthy strains resulting from the framework, synthetically generated damage data are used for training the neuronal network classifier. The synthetic data are obtained by statistical modifications of the healthy strains, to avoid time-consuming and expensive damage simulations. The feature generation framework and health monitoring approach are validated using experiments and numerical simulations of a glass fiber reinforced polymer sandwich structure with a hole considered as damage. The presented numerical and the experimental results clearly show the high potential for the efficient approach for damage detection in a sandwich structure.

Stress-constrained topology optimization of lattice-like structures using component-wise reduced order models

Lattice-like structures can provide a combination of high stiffness with light weight that is useful in many applications, but a resolved finite element mesh of such structures results in a computationally expensive discretization. This computational expense may be particularly burdensome in many-query applications, such as optimization. We develop a stress-constrained topology optimization method for lattice-like structures that uses component-wise reduced order models as a cheap surrogate, providing accurate computation of stress fields while greatly reducing run time relative to a full order model. We demonstrate the ability of our method to produce large reductions in mass while respecting a constraint on the maximum stress in a pair of test problems. The ROM methodology provides a speedup of about 150x in forward solves compared to full order static condensation and provides a relative error of less than 5% in the relaxed stress.