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1. An adaptive weak Galerkin finite element method with hierarchical bases for the elliptic problem

   https://doi.org/10.1002/num.22473  

   Zhang, Jiachuan ; Li, Jingshi ; Li, Jingzhi ; Zhang, Kai ( May 2020 , Numerical Methods for Partial Differential Equations)

   Based on a posteriori error estimator with hierarchical bases, an adaptive weak Galerkin finite element method (WGFEM) is proposed for the elliptic problem with mixed boundary conditions. Forthe posteriorierror estimator, we are only required to solve a linear algebraic system with diagonal entries corresponding to the degree of freedoms, which significantly reduces the computational cost. The upper and lower bounds of the error estimator are shown to addresses the reliability and efficiency of the adaptive approach. Numerical simulations are provided to demonstrate the effectiveness and robustness of the proposed method.

    

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2. A noninvasive method for the determination of in vivo mitral valve leaflet strains

   https://doi.org/10.1002/cnm.3142  

   Rego, Bruno V. ; Khalighi, Amir H. ; Drach, Andrew ; Lai, Eric K. ; Pouch, Alison M. ; Gorman, Robert C. ; Gorman, III, Joseph H. ; Sacks, Michael S. ( September 2018 , International Journal for Numerical Methods in Biomedical Engineering)

   Assessment of mitral valve (MV) function is important in many diagnostic, prognostic, and surgical planning applications for treatment of MV disease. Yet, to date, there are no accepted noninvasive methods for determination of MV leaflet deformation, which is a critical metric of MV function. In this study, we present a novel, completely noninvasive computational method to estimate MV leaflet in‐plane strains from clinical‐quality real‐time three‐dimensional echocardiography (rt‐3DE) images. The images were first segmented to produce meshed medial‐surface leaflet geometries of the open and closed states. To establish material point correspondence between the two states, an image‐based morphing pipeline was implemented within a finite element (FE) modeling framework in which MV closure was simulated by pressurizing the open‐state geometry, and local corrective loads were applied to enforce the actual MV closed shape. This resulted in a complete map of local systolic leaflet membrane strains, obtained from the final FE mesh configuration. To validate the method, we utilized an extant in vitro database of fiducially labeled MVs, imaged in conditions mimicking both the healthy and diseased states. Our method estimated local anisotropic in vivo strains with less than 10% error and proved to be robust to changes in boundary conditions similar to those observed in ischemic MV disease. Next, we applied our methodology to ovine MVs imaged in vivo with rt‐3DE and compared our results to previously published findings of in vivo MV strains in the same type of animal as measured using surgically sutured fiducial marker arrays. In regions encompassed by fiducial markers, we found no significant differences in circumferential(P = 0.240) or radial (P = 0.808) strain estimates between the marker‐based measurements and our novel noninvasive method. This method can thus be used for model validation as well as for studies of MV disease and repair.

    

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3. A convexity enforcing C0 interior penalty method for the Monge–Ampère equation on convex polygonal domains

   Brenner, Susanne C ; Sung, Li-yeng ; Tan, Zhiyu ; Zhang, Hongchao ( June 2021 , Numerische Mathematik)

   We design and analyze a C0 interior penalty method for the approximation of classical solutions of the Dirichlet boundary value problem of the Monge–Ampère equation on convex polygonal domains. The method is based on an enhanced cubic Lagrange finite element that enables the enforcement of the convexity of the approximate solutions. Numerical results that corroborate the a priori and a posteriori error estimates are presented. It is also observed from numerical experiments that this method can capture certain weak solutions. 

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4. A convexity enforcing $${C}^{{0}}$$ interior penalty method for the Monge–Ampère equation on convex polygonal domains

   https://doi.org/10.1007/s00211-021-01210-x  

   Brenner, Susanne C. ; Sung, Li-yeng ; Tan, Zhiyu ; Zhang, Hongchao ( June 2021 , Numerische Mathematik)

   null (Ed.)

   Abstract We design and analyze a $$C^0$$ C 0 interior penalty method for the approximation of classical solutions of the Dirichlet boundary value problem of the Monge–Ampère equation on convex polygonal domains. The method is based on an enhanced cubic Lagrange finite element that enables the enforcement of the convexity of the approximate solutions. Numerical results that corroborate the a priori and a posteriori error estimates are presented. It is also observed from numerical experiments that this method can capture certain weak solutions. 

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5. A mesh‐independent method for planar three‐dimensional crack growth in finite domains

   https://doi.org/10.1002/nme.5946  

   Sheng, A. ; Ghoniem, N. M. ; Crosby, T. ; Po, G. ( September 2018 , International Journal for Numerical Methods in Engineering)

   The discrete crack mechanics (DCM) method is a dislocation‐based crack modeling technique where cracks are constructed using Volterra dislocation loops. The method allows for the natural introduction of displacement discontinuities, avoiding numerically expensive techniques. Mesh dependence in existing computational modeling of crack growth is eliminated by utilizing a superposition procedure. The elastic field of cracks in finite bodies is separated into two parts: the infinite‐medium solution of discrete dislocations and an finite element method solution of a correction problem that satisfies external boundary conditions. In the DCM, a crack is represented by a dislocation array with a fixed outer loop determining the crack tip position encompassing additional concentric loops free to expand or contract. Solving for the equilibrium positions of the inner loops gives the crack shape and stress field. The equation of motion governing the crack tip is developed for quasi‐static growth problems. Convergence and accuracy of the DCM method are verified with two‐ and three‐dimensional problems with well‐known solutions. Crack growth is simulated under load and displacement (rotation) control. In the latter case, a semicircular surface crack in a bent prismatic beam is shown to change shape as it propagates inward, stopping as the imposed rotation is accommodated.

    

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