The Lagrange Multiplier (LM) and penalty methods are commonly used to enforce incompressibility and compressibility in models of cardiac mechanics. In this paper we show how both formulations may be equivalently thought of as a weakly penalized system derived from the statically condensed Perturbed Lagrangian formulation, which may be directly discretized maintaining the simplicity of penalty formulations with the convergence characteristics of LM techniques. A modified Shamanskii–Newton–Raphson scheme is introduced to enhance the nonlinear convergence of the weakly penalized system and, exploiting its equivalence, modifications are developed for the penalty form. Focusing on accuracy, we proceed to study the convergence behavior of these approaches using different interpolation schemes for both a simple test problem and more complex models of cardiac mechanics. Our results illustrate the well-known influence of locking phenomena on the penalty approach (particularly for lower order schemes) and its effect on accuracy for whole-cycle mechanics. Additionally, we verify that direct discretization of the weakly penalized form produces similar convergence behavior to mixed formulations while avoiding the use of an additional variable. Combining a simple structure which allows the solution of computationally challenging problems with good convergence characteristics, the weakly penalized form provides an accurate and efficient alternative to incompressibility and compressibility in cardiac mechanics.

... Alternatively one could use a nearly incompressible formulation, as done in Telle et al. (2021), but with a much higher penalty parameter for the intracellular subdomain. High penalty parameters are, however, associated with locking (Hadjicharalambous et al. 2014;Karabelas et al. 2022), which both can lead to numerical instabilities and underestimation of variables of interest. For these reasons we chose an incompressible formulation. ...

Cardiomyocytes are the functional building blocks of the heart—yet most models developed to simulate cardiac mechanics do not represent the individual cells and their surrounding matrix. Instead, they work on a homogenized tissue level, assuming that cellular and subcellular structures and processes scale uniformly. Here we present a mathematical and numerical framework for exploring tissue-level cardiac mechanics on a microscale given an explicit three-dimensional geometrical representation of cells embedded in a matrix. We defined a mathematical model over such a geometry and parametrized our model using publicly available data from tissue stretching and shearing experiments. We then used the model to explore mechanical differences between the extracellular and the intracellular space. Through sensitivity analysis, we found the stiffness in the extracellular matrix to be most important for the intracellular stress values under contraction. Strain and stress values were observed to follow a normal-tangential pattern concentrated along the membrane, with substantial spatial variations both under contraction and stretching. We also examined how it scales to larger size simulations, considering multicellular domains. Our work extends existing continuum models, providing a new geometrical-based framework for exploring complex cell–cell and cell–matrix interactions.

... In early diastole, residual active stress may be present and in all phases of diastolic filling, the cavity pressure is never zero. Although passive parameter estimates have been shown to be minimally affected by changing the reference state from end-systolic to early-diastolic geometries (Hadjicharalambous et al., 2014b), the impact of the choice of reference state is examined further in Hadjicharalambous et al. (2021) and should be assessed in biventricular patient-specific modelling. ...

Parameterised patient-specific models of the heart enable quantitative analysis of cardiac function as well as estimation of regional stress and intrinsic tissue stiffness. However, the development of personalised models and subsequent simulations have often required lengthy manual setup, from image labelling through to generating the finite element model and assigning boundary conditions. Recently, rapid patient-specific finite element modelling has been made possible through the use of machine learning techniques. In this paper, utilising multiple neural networks for image labelling and detection of valve landmarks, together with streamlined data integration, a pipeline for generating patient-specific biventricular models is applied to clinically-acquired data from a diverse cohort of individuals, including hypertrophic and dilated cardiomyopathy patients and healthy volunteers. Valve motion from tracked landmarks as well as cavity volumes measured from labelled images are used to drive realistic motion and estimate passive tissue stiffness values. The neural networks are shown to accurately label cardiac regions and features for these diverse morphologies. Furthermore, differences in global intrinsic parameters, such as tissue anisotropy and normalised active tension, between groups illustrate respective underlying changes in tissue composition and/or structure as a result of pathology. This study shows the successful application of a generic pipeline for biventricular modelling, incorporating artificial intelligence solutions, within a diverse cohort.

... Biomechanical models were solved using a standard Galerkin FE scheme. Displacement and hydrostatic pressure variables were interpolated using quadratic and linear tetrahedral elements, respectively, to ensure the mixed formulation's stability (Hadjicharalambous et al. 2014b), while quadratic triangular elements were used for epicardial and basal multipliers. All numerical problems were solved using C Heart, a multiphysics FE solver (Lee et al. 2016). ...

A major concern in personalised models of heart mechanics is the unknown zero-pressure domain, a prerequisite for accurately predicting cardiac biomechanics. As the reference configuration cannot be captured by clinical data, studies often employ in-vivo frames which are unlikely to correspond to unloaded geometries. Alternatively, zero-pressure domain is approximated through inverse methodologies, which, however, entail assumptions pertaining to boundary conditions and material parameters. Both approaches are likely to introduce biases in estimated biomechanical properties; nevertheless, quantification of these effects is unattainable without ground-truth data. In this work, we assess the unloaded state influence on model-derived biomechanics, by employing an in-silico modelling framework relying on experimental data on porcine hearts. In-vivo images are used for model personalisation, while in-situ experiments provide a reliable approximation of the reference domain, creating a unique opportunity for a validation study. Personalised whole-cycle cardiac models are developed which employ different reference domains (image-derived, inversely estimated) and are compared against ground-truth model outcomes. Simulations are conducted with varying boundary conditions, to investigate the effect of data-derived constraints on model accuracy. Attention is given to modelling the influence of the ribcage on the epicardium, due to its close proximity to the heart in the porcine anatomy. Our results find merit in both approaches for dealing with the unknown reference domain, but also demonstrate differences in estimated biomechanical quantities such as material parameters, strains and stresses. Notably, they highlight the importance of a boundary condition accounting for the constraining influence of the ribcage, in forward and inverse biomechanical models.

... Several strategies have been proposed in the literature to alleviate the computational burden found in biomechanical simulations. From a theoretical perspective, heart simulations assuming quasi-incompressible and fullyincompressible tissue models problems have been addressed using mixed formulations and FE discretizations [14,17]. In all cases, they have efficiently reduced the computing time by alleviating the volumetric locking phenomenon, a method with a demonstrated history of success in computational mechanics [18,19]. ...

Computational modeling constitutes a powerful tool to understand the biomechanical function of the heart and the aorta. However, the high dimensionality and non-linear nature of current models can be challenging in terms of computational demands. In this work, we present a novel energy-transform variational formulation (ETVF) for accelerating the numerical simulation of hyperelastic biosolids. To this end, we propose a mixed variational framework, where we introduce auxiliary fields that render the strain energy density into a quadratic form, at the expense of adding unknown fields to the problem. We further reduce the non-linearity of the problem by transforming the constraints that arise due to auxiliary fields in a Lagrange multiplier formulation. The resulting continuous problem is solved by using multi-field non-linear finite-element schemes. We assess the numerical performance of the ETVF by solving two benchmark problems in cardiac and vessel mechanics and one anatomically-detailed model of a human heart under passive filling that assumes an orthotropic heterogeneous constitutive relation. Our results show that the ETVF can deliver speed-ups up to 2.28× in realistic cardiovascular simulations only by considering the proposed reformulation of the hyperelastic problem. Further, we show that the ETVF can decrease the wall-clock time of simulations solved in parallel architectures (8-cores) by 55%. We argue that the decrease in computational cost is explained by the ability of the ETVF to reduce the condition number of tangent operators. We believe that the ETVF offers an effective framework to accelerate the numerical solution of general hyperelastic problems, enabling the solution of large-scale problems in attractive computing times. Codes are available for download at https://github.com/dehurtado/ETVF.

... However, the size of the systems involved leads to rather high computational costs, thus making the adoption of robust and efficient iterative solvers important. Apart from the references mentioned above, we stress that within the context of cardiac mechanics, the present contribution complements the works defining strain-based formulations [27], stabilised formulations [28,29], displacement-pressure discontinuous Galerkin [30], and pure displacement [31][32][33][34] methods. In clinically-oriented applications, the performance of numerical solvers is particularly crucial. ...

The numerical approximation of hyperelasticity must address nonlinear constitutive laws, geometric nonlinearities associated with large strains and deformations, the imposition of the incompressibility of the solid, and the solution of large linear systems arising from the discretisation of 3D problems in complex geometries. We adapt the three-field formulation for nearly incompressible hyperelasticity introduced in Chavan et al. (2007) to the fully incompressible case. The mixed formulation is of Hu–Washizu type and it differs from other approaches in that we use the Kirchhoff stress, displacement, and pressure as principal unknowns. We also discuss the solvability of the linearised problem restricted to neo-Hookean materials, illustrating the interplay between the coupling blocks. We construct a family of mixed finite element schemes (with different polynomial degrees) for simplicial meshes and verify its error decay through computational tests. We also propose a new augmented Lagrangian preconditioner that improves convergence properties of iterative solvers. The numerical performance of the family of mixed methods is assessed with benchmark solutions, and the applicability of the formulation is further tested in a model of cardiac biomechanics using orthotropic strain energy densities. The proposed methods are advantageous in terms of physical fidelity (as the Kirchhoff stress can be approximated with arbitrary accuracy and no locking is observed) and convergence (the discretisation and the preconditioners are robust and computationally efficient, and they compare favourably at least with respect to classical displacement–pressure schemes).

... The surface of the wall right behind the beam presents a groove, which we will refer to as sinus. The fluid is modelled as an incompressible Navier-Stokes fluid, while the solid is assumed to behave as an incompressible neo-Hookean material [88]. The main domain characteristics and constitutive law parameters can be found in Table 4. ...

For problems involving large deformations of thin structures, simulating fluid-structure interaction (FSI) remains a computationally expensive endeavour which continues to drive interest in the development of novel approaches. Overlapping domain techniques have been introduced as a way to combine the fluid-solid mesh conformity, seen in moving-mesh methods, without the need for mesh smoothing or re-meshing, which is a core characteristic of fixed mesh approaches. In this work, we introduce a novel overlapping domain method based on a partition of unity approach. Unified function spaces are defined as a weighted sum of fields given on two overlapping meshes. The method is shown to achieve optimal convergence rates and to be stable for steady-state Stokes, Navier-Stokes, and ALE Navier-Stokes problems. Finally, we present results for FSI in the case of 2D flow past an elastic beam simulation. These initial results point to the potential applicability of the method to a wide range of FSI applications, enabling boundary layer refinement and large deformations without the need for re-meshing or user-defined stabilization.

... The deformation was found by solving the linearised total potential energy equations using the CHeart nonlinear mechanics solver [6], using a split u-p formulation outlined in [4]. The reference (X) and deformed geometry (x), and local fiber orientation vectors (f , s, n) fields were interpolated with cubic-Lagrange shape functions, hydrostatic pressure field (p) interpolation was linear Lagrange. ...

Passive material parameter estimation can facilitate the in vivo assessment of myocardial stiffness, an important biomarker for heart failure stratification and screening. Parameter estimation strategies employing biomechanical models of various degrees of complexity have been proposed, usually involving a significant number of cardiac mechanics simulations. The clinical translation of these strategies however is limited by the associated computational cost and the model simplifications. A simpler and arguably more robust alternative is the use of data-based approaches, which do not involve mechanical simulations and can be based for example on the formulation of the energy balance in the myocardium from imaging and pressure data. This approach however requires the estimation of the mechanical work at the myocardial boundaries and the strain energy stored, tasks that are challenging when external loads are unknown - especially at the base which deforms extensively within the cardiac cycle. In this work we employ the principle of virtual work in a strictly data-based approach to uniquely identify myocardial material parameters by eliminating the effect of the unknown boundary tractions at the base. The feasibility of the method is demonstrated on a synthetic data set using a popular transversely isotropic material model followed by a sensitivity analysis to modelling assumptions and data noise.

... Another advantage of using the Kirchhoff stress is that this tensor is symmetric, and, for simpler material laws, is a polynomial function of the displacements (whereas first and second Piola-Kirchhoff stresses are rational functions of displacement) [7]. Alternative remedies for overcoming locking include nonconforming methods (as discussed for the case of cardiac biomechanics in [4]), high order elements and stabilised mixed formulations [56], or Lagrange multiplier-based methods [27]. In our case, solving in terms of stresses proves particularly useful, as this variable participates actively in the electromechanical coupling through the stress-assisted diffusion. ...

We propose and analyse the properties of a new class of models for the electromechanics of the cardiac tissue. The set of governing equations consists of nonlinear elasticity using a viscoelastic and orthotropic exponential constitutive law (this is so for active stress and active strain formulation of active mechanics) coupled with a four-variable phenomenological model for human action potential. The conductivities in the model of electric propagation are modified according to stress, inducing an additional degree of nonlinearity and anisotropy in the coupling mechanisms; and the activation model assumes a simplified stretch-calcium interaction generating active tension and active strain. The influence of the new terms in the electromechanical model is evaluated through a sensitivity analysis, and we provide numerical validation through a set of computational tests using a novel mixed-primal finite element scheme.

Growth and Remodelling (G&R) processes are typical responses to changes in the heart’s loading conditions. The most frequent types of growth in the left ventricle (LV) are thought to involve growth parallel to (eccentric) or perpendicular to (concentric) the fiber direction. However, hypertrophic cardiomyopathy (HCM), a genetic mutation of the sarcomeric proteins, exhibits heterogeneous patterns of growth and fiber disarray despite the absence of clear changes in loading conditions. Previous studies have predicted cardiac growth due to increased overload in the heart [7, 12, 23] as well as modelled inverse G&R post-treatment [1, 14]. Since observed growth patterns in HCM are more complex than standard models of hypertrophy in the heart, fewer studies focus on the geometric changes in this pathological case. By adapting established kinematic growth tensors for the standard types of hypertrophy in an isotropic and orthotropic material model, the paper aims to identify different factors which contribute to the heterogeneous growth patterns observed in HCM. Consequently, it was possible to distinguish that fiber disarray alone does not appear to induce the typical phenotypes of HCM. Instead, it appears that an underlying trigger for growth in HCM might be a consequence of factors stimulating isotropic growth (e.g., inflammation). Additionally, morphological changes in the septal region resulted in higher amounts of incompatibility, evidenced by increased residual stresses in the grown region.

Background
Myocardial volume is assumed to be constant over the cardiac cycle in the echocardiographic models used by professional guidelines, despite evidence that suggests otherwise. The aim of this paper is to use literature-derived myocardial strain values from healthy patients to determine if myocardial volume changes during the cardiac cycle.
Methods
A systematic review for studies with longitudinal, radial, and circumferential strain from echocardiography in healthy volunteers ultimately yielded 16 studies, corresponding to 2917 patients. Myocardial volume in systole (MVs) and diastole (MVd) was used to calculate MVs/MVd for each study by applying this published strain data to three models: the standard ellipsoid geometric model, a thin-apex geometric model, and a strain-volume ratio.
Results
MVs/MVd<1 in 14 of the 16 studies, when computed using these three models. A sensitivity analysis of the two geometric models was performed by varying the dimensions of the ellipsoid and calculating MVs/MVd. This demonstrated little variability in MVs/MVd, suggesting that strain values were the primary determinant of MVs/MVd rather than the geometric model used. Another sensitivity analysis using the 97.5th percentile of each orthogonal strain demonstrated that even with extreme values, in the largest two studies of healthy populations, the calculated MVs/MVd was <1.
Conclusions
Healthy human myocardium appears to decrease in volume during systole. This is seen in MRI studies and is clinically relevant, but this study demonstrates that this characteristic was also present but unrecognized in the existing echocardiography literature.

Degrees of freedom which are Lagrange multipliers arise in the finite element approximation of mixed variational problems. When these degrees of freedom are ″local″ , the introduction of a small perturbation (corresponding by duality to a penalty function) enables the elimination of these unknowns at the element level. This method is examined for the continuous case. It is shown that the solution of the perturbed problem is close to that of the original one. This result is extended to the FEM. Several examples are given and the construction of a number of the element stiffness matrices is outlined.

In this thesis, an anatomically accurate finite element model of the left ventricle is presented for coupled fluid- solid simulation of blood flow and tissue mechanics during passive filling. Beginning from continuum theory, general conservation principles – common to both blood flow and tissue mechanics – are derived in the ALE frame. The deviation of these underlying principles for fluid and solid mechanics is then discussed, and their respective weak forms shown. A coupling technique is then devised which allows non-conformity of the fluid- solid problems by introducing an additional weak constraint. It is then proven, for a linear fluid-solid system, that this coupling strategy produces unique stable solutions which, in the discrete setting, converge optimally (under certain restrictions on the solution’s smoothness). The methods discussed are subsequently implemented into a parallel software framework, and extensively verified using numerical experiments. The culmination of these efforts enables the creation of a left ventricular heart model (which is shown to have bounded a priori energy estimates), incorporating known information on tissue architecture and material properties. This model is then used to conduct a comprehensive energy analysis of passive filling in the model left ventricle.

A mixed variational theorem for linear orthotropic thermoelastic solids is presented. The mechanical state variables are taken to be the displacement vector and a scalar stress variable. The Euler equations of the variational principle are the displacement equations of equilibrium and a condition relating the stress variable to strain and temperature change. An important feature of the principle is that the field equations for both compressible and incompressible solids may be generated. In connection with applications to the development of finite element computer algorithms for the solution of boundary value problems a well-conditioned system of equations is obtained for nearly-incompressible solids.

Designing engineering components that make optimal use of materials requires consideration of the nonlinear characteristics associated with both manufacturing and working environments. The modeling of these characteristics can only be done through numerical formulation and simulation, and this requires an understanding of both the theoretical background and associated computer solution techniques. By presenting both nonlinear continuum analysis and associated finite element techniques under one roof, Bonet and Wood provide, in this edition of this successful text, a complete, clear, and unified treatment of these important subjects. New chapters dealing with hyperelastic plastic behavior are included, and the authors have thoroughly updated the FLagSHyP program, freely accessible at www.flagshyp.com. Worked examples and exercises complete each chapter, making the text an essential resource for postgraduates studying nonlinear continuum mechanics. It is also ideal for those in industry requiring an appreciation of the way in which their computer simulation programs work.

The paper is devoted to a general finite element approximation of the solution of the Stokes equations for an incompressible viscous fluid. Both conforming and nonconforming finite-element methods are studied and various examples of simplistic elements well suited for the numerical treatment of the incompressibility condition are given. Optimal error estimates are derived in the energy norm and in the L**2-norm.

A variational formulation for finite isothermal deformations of isotropic hyperelastic materials is presented. This is equivalent to nonlinear elastic field (Lagrangean) equations expressed in terms of the displacement field and a scalar function that is associated with the hydrostatic mean stress when specialized to classical elasticity. The variational formulation is particularly useful in the development of finite element analysis of nearly incompressible and incompressible materials. For slightly compressible materials small volume changes are not assumed in our model. The formulation gives a reasonable transition of results for materials varying from compressible to incompressible and is general in the sense that it admits a general form of constitutive equation. The formulation for incompressible materials is recovered from the compressible one simply as a limit. It can be considered as an extension of Herman's principle to nonlinear elasticity. Several numerical simulations are presented that show the performance of the proposed formulation and the convergence behavior of different types of elements for compressible and incompressible elasticity.