Comparisons of implicit and explicit time integration methods

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Comparisons of implicit and explicit time integration methods

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COMPARISONS OF IMPLICIT AND EXPLICIT TIME INTEGRATION METHODS IN FINITE ELEMENT ANALYSIS FOR LINEAR ELASTIC
MATERIAL AND QUASI-BRITTLE MATERIAL IN DYNAMIC PROBLEMS
A thesis submitted to the Delft University of Technology to fulfill the requirements for the degree of
Master of Science in Structural Engineering (Structural Mechanics)
by Boyuan Yang October 2019

Boyuan Yang: Comparisons of implicit and explicit time integration methods in finite element analysis for linear elastic material and quasi-brittle material in dynamic problems (2019)

The work in this thesis was made in the:

DIANA FEA & Faculty of Civil Engineering and Geosciences Delft University of Technology

Project duration: Thesis committee:

January, 2019 - October, 2019 Dr. ir. M.A.N. Hendriks, Dr. ir. G.M.A. Schreppers, Prof. dr. ir. J.G. Rots, Dr. ir. A. Tsouvalas,

TU Delft, chairman DIANA FEA TU Delft TU Delft

ABSTRACT
In finite element analysis, nonlinear time-history analysis is a realistic and accurate analysis type for dynamic or seismic analysis due to its solutions contain wealthy data and complete response time-history. The most commonly used method, probably the only practical procedure, in nonlinear time-history analysis is the direct time integration method. It solves the governing equations of the system in time domain incrementally. In general, every direct time integration method could be classified as either an implicit method or an explicit method. Each category has its advantages and disadvantages in different aspects, e.g., stability, accuracy and computational costs. Understanding the differences between the two categories in both theoretical and practical aspects is very important for engineers to make the best analysis strategy for a specific dynamic or seismic analysis.
In this treatise, the fundamental theory of the direct time integration methods and several well-known methods will be reviewed. Many published comparisons, either in theoretical or practical level, will be briefly covered. Then, the most popular method in each category, i.e., implicit Newmark method and explicit central difference method, will be introduced and used in transient analyses and results comparisons. In total, five cases studies are included in this thesis, including three cases with linear elastic materials and two cases with quasi-brittle masonry material. These five cases are studied to answer the main research questions of this research:
What differences can be observed in comparisons of solutions obtained from implicit and explicit methods for linear elastic material in transient analysis and for quasi-brittle material under seismic load? Also, how are the performances of both methods with respect to the stability and accuracy aspects?
The finite element models of all cases are built up in DIANA FEA 10.3, and transient analyses with both implicit and explicit methods are performed as well.
The first three cases with linear elastic materials include a simply supported beam under a harmonic point load, a double cantilever beam under point transient load ,and a simply supported thin plate under transient distributed load. The remaining two cases with quasi-brittle masonry material are the seismic analyses of a masonry wall model and a full-scaled URM house model (finally simplified), and they are referred from the experimental tests conducted by Graziotti et al. [2016] and modified in this thesis. Different analyses schemes are set for each case in order to investigate the influence of the adopted time step in each method. Finally, the comparisons are made between implicit and explicit method solutions concerning displacement responses for linear elastic material, and additionally cracks patterns and capacity curves for masonry material.
Based on the comparisons, the conclusions can be drawn to answer the main research questions.
For linear elastic materials
• The results show that both methods generally could accurately reproduce the displacement responses with proper time step. Few differences are observed in the displacement or stress responses of high frequency contents. The implicit method was strongly influenced by the adopted time step to ensure the accuracy of the high frequency vibration responses. Though the implicit method is unconditionally stable, a large time step could make high frequency information lost in the solution.
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• The explicit method once satisfies the stability condition (called CFL stability condition), which means the time step used for the algorithm to proceed is smaller than the minimum natural period of all elements (called critical time step), the high frequency responses will always be accurately calculated, with regardless of large output time intervals. Moreover, the accuracy and stability could both be guaranteed once the CFL condition is satisfied, which means further decrement in time step is not necessary in the explicit method. However, since the critical time step is usually very small, rather long computation time is needed.
For quasi-brittle masonry material
• The comparisons show that the implicit and explicit solutions generally have a good agreement with each other in terms of displacement response, hysteresis curves, and crack patterns. The critical time step, determined by CFL based on the linear elastic phase of structure, could guarantee the accuracy and also stability for the material, which has a softening behavior, e.g., quasibrittle masonry material. Similarly, no further reduction for critical time step is needed.
• The implicit method shows some difficulties to reach the convergence, as a result, there are some chaotic results in hysteresis curves of displacement versus base shear. Non-converged or hardly converged iteration procedures in implicit method could lead to inaccurate predictions of nonlinear behaviors.
• The explicit method has good results with smooth transitions in hysteresis curves, which benefit from no iteration involving. This advantage of the explicit method could be more significant when highly nonlinear behaviors are involved in the analysis. However, the disadvantage is that the explicit method needs a very small time step. For a complex model, the critical time step will be extremely reduced due to irregular-shaped mesh and connections with volumes close to zero.
• Mass scaling technique could be used to speed up the explicit method by adding artificial mass on specific elements to increase the available critical time step for the whole FE model. However, great caution is needed to use this technique. Generally speaking, the ratio between added mass and total mass should smaller than 10%. Slightly larger values could be allowed only with a detailed check of positions and properties of the elements with added mass.
According to the conclusions, the explicit method should be preferable in either one or several of the following situations:
• Short duration transient analysis, e.g., impact loading analysis.
• The high frequency vibrations are of interest, e.g., seismic analysis of high-rise buildings.
• Highly nonlinear behaviors are included in the model, which may cause enormous difficulties for convergence of the iteration process, e.g., severe cracking, local failure or crushing.
• The target structure has a regular geometry and a well-meshed FE model.
For other situations, the implicit method should be recommended due to its relatively large time step and unconditional stability.

ACKNOWLEDGEMENTS
First, I am very grateful to my company supervisor Gerd-Jan Schreppers, who gave me the opportunity to work on this topic and in DIANA FEA BV. He patiently taught me on learning finite element method and using DIANA FEA. Also, I would like to express many thanks to Tanvir Rahman, Angelo Garofano, Kesio Palacio, and Arno Wolthers, they generously provided me lots of help whenever I needed. Specially, thanks to my friend Manimaran Pari, who sits next to me and gave me precious advice on my work and thesis writing. I am really grateful to performing my graduation project in DIANA FEA BV, everyone in this company is so kind and generous. It is my great pleasure to work with all of you during the past 9 months.
Also, many thanks to Max Hendriks, Jan Rots and Apostolos Tsouvalas to join my committee. They gave me valuable guidance and suggestions to my work on every meeting. The discussions with them greatly helped me to form this thesis.
Thanks to all of my friends in this beautiful place. Every moment spent with you is always enjoyable to me.
Finally, I would like to give my deepest gratitude to my parents and my elder sister, they support my study and life in TU Delft, and always give me courage to move forward. Above all, Yiting, who is always there whenever I needed. Her accompany is the most encouragement for me in these years.
Delft, October 2019
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CONTENTS

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1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Research questions and scope . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

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2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Nonlinear dynamic time-history analysis . . . . . . . . . . . . . . . . . 5
2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 Classification of direct time integration methods . . . . . . . . 6 2.3 Widely used direct time integration methods . . . . . . . . . . . . . . . 6 2.3.1 The Newmark Method . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.2 The Wilson θ Method . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.3 The Houbolt Method . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.4 The Central Difference Method . . . . . . . . . . . . . . . . . . . 8 2.4 Review of comparisons made between implicit and explicit finite element methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.1 Stability and Accuracy of direct integration methods . . . . . . 9 2.4.2 Comparisons in several practical problems . . . . . . . . . . . . 10 2.5 Recent Development of the direct time integration . . . . . . . . . . . . 11

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3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Implicit Newmark method . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2.1 Step-by-step solution procedure . . . . . . . . . . . . . . . . . . 14 3.2.2 The implicit integration of nonlinear equations in dynamic
analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Explicit central difference method . . . . . . . . . . . . . . . . . . . . . 16
3.3.1 Step-by-step solution procedure . . . . . . . . . . . . . . . . . . 17 3.3.2 The critical time step and CFL stability condition . . . . . . . . 17 3.3.3 The explicit integration of nonlinear equations in dynamic
analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.4 Important features of direct time integration methods in DIANA FEA
10.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.4.1 Numerical damping in Newmark method . . . . . . . . . . . . 19 3.4.2 Iteration method and convergence criteria . . . . . . . . . . . . 20 3.4.3 Mass and damping matrices . . . . . . . . . . . . . . . . . . . . 20 3.4.4 Start procedure of explicit method and time step definition . . 20 3.4.5 Limitation of element order and mass scaling in explicit method 21 3.4.6 Stability control in explicit method: energy balance . . . . . . . 22

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23

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 Case 1: A simply-supported beam subjected to a harmonic point load 23

4.2.1 Case description . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.2 Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.3 Analyses schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3 Case 2: A double cantilever beam subjected to a transient point load . 26

4.3.1 Case description . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3.2 Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . 27

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4.3.3 Analyses schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.4 Case 3: A simply-supported thin plate under out-of-plane transient
distributed load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.4.1 Case description . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.4.2 Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.4.3 Analysis schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.5 Case 4: The in-plane loading tests of masonry wall EC-COMP2-3 . . 31 4.5.1 Case description . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.5.2 In-plane shear-compression test . . . . . . . . . . . . . . . . . . 32 4.5.3 In-plane seismic analysis . . . . . . . . . . . . . . . . . . . . . . 36 4.6 Case 5: The URM full-scale building tests . . . . . . . . . . . . . . . . . 41 4.6.1 Case description . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.6.2 Geometry and general characteristics of the house . . . . . . . 41 4.6.3 Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.6.4 Input seismic signal . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.6.5 Preliminary analyses and results . . . . . . . . . . . . . . . . . . 48 4.6.6 Simplified model . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2.1 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2.2 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2.3 Comparison of the solutions . . . . . . . . . . . . . . . . . . . . 57

5.3 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3.1 Sub-case 1: solutions of time step 25µs . . . . . . . . . . . . . . 58

5.3.2 Sub-case 2: solutions of time step 50µs . . . . . . . . . . . . . . 58

5.3.3 Sub-case 3: solutions of time step 100µs . . . . . . . . . . . . . 59

5.3.4 Comparisons of solution from different time steps . . . . . . . 59

5.4 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.4.1 Eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4.2 Displacement time history in out-of-plane direction . . . . . . . 61

5.4.3 Stress σxx time history . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4.4 Comparisons and discussions . . . . . . . . . . . . . . . . . . . . 62

5.5 Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.5.1 In-plane shear-compression test . . . . . . . . . . . . . . . . . . 64

5.5.2 In-plane seismic analysis . . . . . . . . . . . . . . . . . . . . . . 67

5.5.3 Comparisons and discussions . . . . . . . . . . . . . . . . . . . . 69

5.5.4 Additional analysis: using the explicit critical time step in im-

plicit method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.6 Case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.6.1 Eigenfrequency analysis . . . . . . . . . . . . . . . . . . . . . . . 73

5.6.2 Added mass ratios . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.6.3 Relative displacement response at first floor . . . . . . . . . . . 74

5.6.4 Hysteresis curves of base shear versus first floor average dis-

placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.6.5 Deformed shapes and crack patterns . . . . . . . . . . . . . . . 77

5.6.6 Comparisons and discussions . . . . . . . . . . . . . . . . . . . . 78

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6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

References

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MethodTime StepAnalysisComparisonsTime Integration Methods